Electricity

ELECTRICITY

THE word Electricity is derived from the Greek word ηλeκτpov, meaning amber. The term was invented by Gilbert, who used it with reference to the attractions and repulsions excited by friction in certain bodies of which amber may be taken as the type. To the cause of these forces was given the name Electricity; and out of the study of these and kindred phenomena arose the science of electricity, of which it is the purpose of the present article to give a brief outline.

The science has been divided into three branches— Electrostatics, which deals with electricity at rest; Electrokinetics, which considers the passage of electricity from place to place; and Electromagnetism, which treats of the relation of electricity to magnetism. We shall, however, make no attempt to adhere to this division, but shall exhibit the different parts of the subject in such order and connection as seems most clear and natural in the present state of the science. For the sake of the non-scientific reader we prefix a brief history of the science of electricity, wherein mention is made of some of the more striking electrical discoveries and of the steps by which our knowledge of the subject has advanced to its present condition.

Historical Sketch.

The name of the philosopher who first observed that amber when rubbed possesses the property of attracting and repelling light bodies has not been handed down to our times. Thales of Miletus is said to have described this remarkable property, and both Theophrastus (321 β.c.) aud Pliny (70 a.d.) mention the power of amber to attract straws and dry leaves. The same authors speak of the lapis lyncumus, which is supposed to be a mineral called tourmaline, as possessing the same property. The electricity of the torpedo was also known to the ancients. Pliny informs us, that when touched by a spear it paralyzes the muscles and arrests the feet, however swift; and Aristotle adds that it possesses the power of benumbing men, as well as the fishes which serve for its prey. The influence of electricity on the human body, and the electricity of the human body itself, were also known in ancient times. Anthero, a freedman of Tiberius, was cured of the gout by the shocks of the torpedo; and Wolimer, the king of the Goths, was able to emit sparks from his own body. Eustathius, who records this fact, also states that a certain philosopher, while dressing and undressing, emitted occasionally sudden crackling sparks, while at other times flame3 blazed from him without burning his clothes. Such are the scanty gleanings of electrical knowledge which we derive from the ancient philosophy; and though several writers of the Middle Ages have made occasional references to these facts, and even attempted to speculate upon them, yet they added nothing to the science, and left an open field for the researches of modern philosophers.

Dr Gilbert of Colchester may be considered as the founder of the science, as he appears to have been the first philosopher who carefully repeated the observations of the ancients, and applied to them the principles of philosophical investigation. In order to determine if other bodies possessed the same property as amber, he balanced a light metallic needle on a pivot, and observed whether or not it was affected by causing the excited or rubbed body to approach to it. In this way he discovered that the following bodies possess the property of attracting light substances :— amber, gagates or jet, diamond, sapphire, carbuncle, rock-crystal, opal, amethyst, vincentina or Bristol stone, beryl, glass, paste for false gems, glass of antimony, slags, belemnites, sulphur, gum-mastic, sealing-wax of lac, hard resin, arsenic, rock salt, mica, and alum. These various bodies attracted, with different degrees of force, not only straws and light films, but likewise metals, stones, earths, wood, leaves, thick smoke, and all solid and fluid bodies. Among the substances which are not excited by friction Gilbert enumerated emerald, agate, carnelian, j>earls, jasper, calcedony, alabaster, porphyry, coral, marble, Lydian stone, flints, hematites, smyris (emery or corundum), bones, ivory, hard woods, such as cedar, ebony, juniper, and cypress, metals, and natural magnets. Gilbert also discovered that the state of the atmosphere affects the production of electricity; dryness with north or east wind being a favourable condition, while moisture with south wind is unfavourable. An account of Gilbert’s experiments will be found in his book De Magnete, lib. ii. cap. 2.

Robert Boyle added many new facts to the science of electricity, and he has given a full account of them in his Experiments on the Origin of Electricity. By means of a suspended needle, he discovered that amber retained its attractive virtue after the friction which excited it had ceased; and though smoothness of surface had been regarded as advantageous for excitation, yet he found a diamond which in its rough state exceeded all the polished ones and all the electrics which he had tried, having been able to move a needle three minutes after he had ceased to rub it. He found also that heat and tersion (or the cleaning or wiping of any body) increased its susceptibility of excitation; and that if the attracted body were fixed, and the attracting body movable, their mutual approach would still take place. To Gilbert’s list of “electrics” Boyle added the resinous cake which remained after evaporating one-fourth part of good oil of turpentine, the dry mass which remains after distilling a mixture of petroleum and strong spirit of nitre, glass of lead, caput mortuum of amber, white sapphire, white amethyst, diaphanous ore of lead, carnelian, and a green stone supposed to be a sapphire.

To these discoveries of Boyle his contemporary Otto 1 von Guericke added the highly important one of electric light (Experimenta Nova Magdeburgica, lib. iv. cap. 15). Having cast a globe of sulphur in a glass sphere, and broken off the glass, he mounted the sulphur ball upon a revolving axis, and excited it by the friction of the hand. By this means he discovered that light and sound accompanied strong electrical excitation, and he compares the light to that which is exhibited by breaking lump sugar in the dark. With this powerful apparatus Guericke verified on a greater scale the results obtained by his predecessors, and obtained several new ones of very considerable importance. He found that a light body, when once attracted by an excited electric, was repelled by it, and was incapable of a second attraction until it had been touched by some other body; and that light bodies suspended within the sphere of influence of an excited electric possessed the same properties as if they had been excited.

To our illustrious countryman Sir Isaac Newton the science of electricity owes some important observations. He used in his electrical experiments a globe of glass rubbed by the hand instead of the sulphur globe of Von Guericke. It would appear that Newton was the first to use glass in this way (Optics, query 8th). We owe also to Sir Isaac a beautiful experiment on the excitation of electricity which has since become very popular. Having fixed a round disc of glass in a short brass cylinder, he placed small pieces of thin paper within the cylinder and upon a table, so that the lower surface of the glass was one-eighth of an inch distant from the table. He then rubbed the upper surface of the glass, and he observed the pieces of paper “leap from one part of the glass to the other, and twirl about in the air.” This experiment, after a previous unsuccessful trial, was repeated by the Royal Society in 1676 (Brewster’s Life of Neιvton, p. 307).

Francis Hawksbee, one of the most active experimental philosophers of his age, added many new facts to the science. In 1705 he communicated to the Royal Society several curious experiments on what he calls “the mercurial phosphorus.” He showed that light could be produced by passing common air through mercury placed in a well-exhausted receiver. The air rushing through the mercury, blew it up against the sides of the glass that held it, “appearing all around like a body of fire, consisting of abundance of glowing globules.” The phenomenon continued till the receiver was half full of air. These phenomena had been observed in the Torricellian vacuum before Hawksbee’s time, and various explanations suggested. He suspected that they were due to electricity, and remarked their resemblance to lightning. Like Newton he used a revolving glass globe rubbed by the hand to generate electricity. Besides the experiment above alluded to he made many others on the electric light and on the attractions of electrified bodies. Descriptions of these will be found in his Physico-Mechanical Experiments, 1709, and in several memoirs in the Philosophical Transactions about 1707.

About the same time Dr Wall (Phü. Trans., 1708) observed the spark and crackling sound accompanying the electrical excitation of amber, and compared them to thunder and lightning.

One of the most ardent experimentalists of his time was Stephen Gray, a Fellow of the Royal Society. In his first paper, published in 1720, he showed that electricity could be excited by the friction of feathers, hair, silk, linen, woollen, paper, leather, wood, parchment, and gold-beaters’ skin. Several of these bodies exhibited light in the dark, especially after they had been warmed; but all of them attracted light bodies, and sometimes at the distance of eight or ten inches. An epoch was made in the history of electricity by the discovery of Gray in 1729, that certain bodies had, while others had not, the power of conveying electricity from one body to another, i.e., in modern phrase, conducting it. Gray experimented with a glass tube, into the ends of which were fastened two corks; into one of these he fastened a fir rod, and to the end of the rod an ivory ball. On rubbing the glass he found that'the ball attracted the light bodies as vigorously as the glass itself. He made a variety of experiments with rods of different length, and with a packthread, by which he suspended his ball from the balcony of an upper story of his house, all with the same result. He then attempted to carry the electricity horizontally on a packthread which he suspended with hempen strings; but the experiment failed. On the occasion of a repetition of the experiments at the house of his friend Wheeler, silk strings were suggested as a support, and found to answer, while metal wires failed. Gray and Wheeler were thus led to the conclusion that it was the material of the supports that was in question, and that whereas packthread had, silk had not the power of transmitting electricity to a distance. Gray and Wheeler managed, by supporting a packthread by silk loops, to convey electricity from a piece of rubbed glass to a distance of 886 feet. The conducting power of fluids, and of the human body, was established by Gray. He also made many curious experiments on the electrical properties of resinous cakes, which he allowed to cool and harden in the ladles in which they had been melted. For an account of these and others the student is referred to memoirs in the Philosophical Transactions for 1731, 1735, Ac.

Desaguliers made many experiments confirming Gray’s conclusions, and found that bodies that have the property of being electrically excitable by friction, or electrics per se, have not the pow er of conduction; whereas conductors are not electrics per se. These terms, introduced by him, were useful in bringing into concise and scientific language the discoveries of Gray.

While Gray was pursuing his career of discovery in England, M. Dufay, of the Academy of Sciences, and superintendent of the Royal Botanic Gardens, was actively employed in the same researches. He found that all bodies, whether solid or fluid, could be electrified by an excited tube, by setting them on a glass stand slightly warmed, or only dried; and that those bodies which are in themselves least electrical received the greatest degree of electricity from the approach of the glass tube. He repeated the experiments of Gray, confirming his results, and found that electricity was transmitted more easily along packthread when it was wetted, and that it might be supported upon glass tubes in place of silk lines. In this way he conveyed it along a string 1256 feet long. He suspended by silken strings and electrified a child as Gray had done; and having suspended himself in a similar manner, he discovered that an electrical spark, accompanied with a crackling noise, took place when any other person touched him, and he has described the prickling sensation like the burning from a spark of fire, which is at the same time felt either through the clothes or on the skin. The great discovery of Dufay, however, was that of two different kinds of electricity. He fully recognized the importance of this fundamental fact, and gave the name of vitreous electricity to that which is produced by exciting glass, rock-crystal, precious stones, hair of animals, wool, and many other bodies; and the name of resinous to that which is produced by exciting resinous bodies, such as amber, copal, gum-lac, silk, paper, thread, and a number of other substances. The characteristic of those two electricities was, that a body with vitreous electricity attracted all bodies with resinous electricity, and repelled all bodies with vitreous electricity; while a body with resinous electricity attracted all bodies with vitreous electricity, aud repelled all bodies with resinous electricity. Two electrified silk threads, for example, repel each other, aud also two electrified woollen threads, but an electrified silk thread will attract an electrified woollen thread. Hence it is easy to determine whether any body possesses vitreous or resinous electricity. If it attracts an electrified silk thread, its electricity will be vitreous; if it repels it, it will be resinous.

Gray repeated and varied the experiments of Dufay, and made mauy new ones. Like Hawksbee and Dr Wall, he recognized the similarity between the phenomena of electricity and those of thunder and lightning; and he expresses a hope “that there may be found out a way to collect a greater quantity of electric fire, and consequently to increase the force of that power, which, by several of these experiments, si licet magwis componere parva, seems to be of the same nature with thunder and lightning.”

The discoveries which we have now recounted began to rouse the activity of the German and Dutch philosophers. To the electrical machine used by Newton and Hawksbee, Professor Boze of Wittenberg added the prime conductor, which at first consisted of an iron or tin tube supported by a man standing upon cakes of rosin; but it was afterwards suspended by silken strings. Professor Winkler of Leipsic substituted a cushion in place of the hand for exciting the revolving globe; and Professor Gordon of Erfurt, a Scotch Benedictine monk, first used a glass cylinder, eight inches long and four broad, which he caused to revolve by means of a bow and string. By these means electrical sparks of great size and intensity were produced, and by their aid various combustible substances, both fluid and solid, were inflamed. In 1744 M. Ludolph of Berlin succeeded in firing, by the electrical spark, the ethereal spirit of Frobenius. Winkler did the same by a spark from his finger; and he succeeded in inflaming French brandy and other weaker spirits after they had been heated. Gordon kindled spirits by a jet of electrified water. Dr Miles inflamed phosphorus by the electric spark; and oil, pitch, and sealing-wax, when strongly heated, were set on fire by similar means. We refer the student for lists of the works of the philosophers just mentioned to the admirable bibliography given by Young, Natural Philosophy, p. 515.

These striking effects were all produced by the electricity obtained immediately from an excited electric; but a great step was now made in the science by the discovery of a method of accumulating and preserving electricity in large quantities. The author of this great invention is not distinctly known; but there is reason to believe that a monk of the name of Kleist, a person of the name of Cuneus, and Professor Muschenbroeck of Leyden had each the merit of an independent inventor. The invention by which this accumulation was effected was called the Legden Jar or Phial, because it was principally in Leyden that it was either invented or tried. Having observed that excited electrics soon lost their electricity in the open air, and that their loss was accelerated when the atmosphere was charged with moisture or other conducting materials, Muschenbroeck conceived that the electricity of bodies might be retained by surrounding them with bodies which did not conduct it. In putting this idea to the test of experiment, he electrified some water in a glass bottle, and a communication having been made between the water and the prime conductor, the assistant, who was holding the bottle, on trying to disengage the communicating wire, received a sudden shock in his arms and breast, aud thus established the efficacy of the Leyden jar.

Sir William Watson made some important experiments at this period of our history (Memoirs in Phil. Trans. about 1747). He succeeded in firing gunpowder by the electric spark; and by mixing the gunpowder with a little camphor he discharged a musket by the same power. He also fired hydrogen by the electric spark; and he kindled both spirits of wine and hydrogen by means of a drop of cold water, and even with ice. In the German experiments the fluid or solid to be inflamed was set on fire by an electrified body; but Sir William Watson placed the fluid in the hands of an electrified person, and set it on fire by causing a person not electrified to touch it with his finger. Sir William Watson first observed the flash of light which attends the discharge of the Leyden phial, and it is to him that we owe the present improved form of the Leyden phial, in which it is coated both without and within with tinfoil. Dr Bevis indeed had suggested the outside coating, and at Smeaton’s recommendation, he coated a pane of glass on both sides, and within an inch of the edge, with tinfoil; but still the idea of coating the jar doubly belongs to Sir William Watson.

A party of the Royal Society, with the president at their head, and Sir William Watson as their chief operator, entered upon a series of magnificent experiments, for the purpose of determining the velocity of the electric fluid, and the distance to which it could be conveyed. The French savans had conveyed the influence of the Leyden jar through a circuit of 12,000 feet; and in one case the basin at the Tuileries, containing about an acre of water, formed part of the circuit; but the English philosophers made a more complete series of experiments, of which the following were the results :—

1. That in all their operations, when the wires have been properly conducted, the electrical commotions from the charged phial have been very considerable only when the observers at the extremities of the wire have touched some substance readily conducting electricity with some part of their bodies.

2. That the electrical commotion is always felt most sensibly in those parts of the bodies of the observers which are between the conducting wires and the nearest and the most non-electric substance, or, in other words, so much of their bodies as comes within the electrical circuit.

3. That on these considerations we infer that the electrical power is conducted between these observers by any non-electric substances which happen to be situated between them, and contribute to form the electrical circuit.

4. That the electrical commotion has been perceptible to two or more observers at considerable distances from each other, even as far as two miles.

5. That when the observers have been shocked at the end of tws∙ miles of wire, we infer that the electrical circuit is four miles, viz. two miles of wire, and the space of two miles of the non-electric matter between the observers, whether it be water, earth, or both.

6. That the electrical commotion is equally strong, whether it is conducted by water or dry ground.

7. That if the wires between the electrifying machine and the observers are conducted on dry sticks, or other substances nonelectric in a slight degree only, the effects of the electrical power are much greater than when the wires in their progress touch the ground, or moist vegetables, or other substances in a great degree non-electric.

8. That by comparing the respective velocities of electricity and sound, that of electricity, in any of the distances yet experienced, is nearly instantaneous.

In the following year these experiments were resumed with the view of ascertaining the absolute velocity of electricity at a certain distance, and it was found “that through the whole length of a wire 12,276 feet the velocity of electricity was instantaneous.”

The theory of positive and negative electricity which was afterwards elaborated by Franklin, was distinctly announced by Sir W. Watson. He lays it down as a law that in electrical operations there is an afflux of “electric fluid” to the globe and the conductor, and also an efflux of the same matter from them. In the case of two insulated persons, the one in contact with the rubber and the other with the conductor, he observed that either of them would communicate a much stronger spark to the other than to any bystander. The electricity of the one, he says, became more rare than it is naturally, and that of the other more dense, so that the density of the electricity in the two insulated persons differed more than that between either of them and a bystander.

A variety of interesting experiments were made about this time by Le Monnier, Nollet, Winckler, Ellicott, Jallabert, Boze, Menon, Smeaton, and Miles. In 1746 Le Monnier confirmed the result previously obtained by Gray, that electricity is communicated to homogeneous bodies in proportion to their surfaces only. Boze discovered that capillary tubes which discharged water by drops afforded a continuous stream when electrified. The Abbé Nollet {Essai sur l'Electricite, 1746; Recherches, 1749; Lettres, 1753), the friend and coadjutor of Dufay, ascertained that electricity increases the natural evaporation of fluids, and that the evaporation is hastened by placing them in non-electric vessels. Jallabert confirmed the result previously obtained by Watson, that electricity passes through the substance of a conducting wire, and not along its surface. Smeaton found that the red hot part of an iron bar could be as strongly electrified as the cold parts on each side of it. Dr Miles kindled common spirits by a stick of black sealing-wax excited by dry flannel. Ellicott conceived that the particles of the electric fluid repel each other, while they attract those of all other bodies. Mowbray concluded that the vegetation of two myrtles was hastened by electrifying them,— a result which Nollet confirmed in the case of vegetating seeds. The Abbé Menon found that cats, pigeons, sparrows, and chaffinches lost weight by being electrified for five or six hours, and that the same result was true of the human body; and hence it was concluded that electricity augments the insensible perspiration of animals.

A high place in the history of electricity must be allotted to the name of Dr Benjamin Franklin of Philadelphia. His researches did much to extend our theoretical and practical knowledge of electricity, and the clearness and vigour of his style made his writings popular, and spread the study of the subject.

One of the first labours of the American philosopher was to present, in a more distinct form, the theory of positive and negative electricity, which Sir W. Watson had been the first to suggest. He showed that electricity is not created by friction, but merely collected from its state of diffusion through other matter by which it is attracted. He asserted that the glass globe, when rubbed, attracted the electrical fire, and took it from the rubber, the same globe being disposed, when the friction ceases, to give out its electricity to any body which has less. In the case of the charged Leyden jar, the inner coating of tinfoil had received more than its ordinary quantity of electricity, and was therefore electrified positively or phis, while the outer coating of tinfoil having had its ordinary quantity of electricity diminished, was electrified negatively or minus. Hence the cause of the shock and spark when the jar is discharged, or when the superabundant plus electricity of the inside is transferred by a conducting body to the defective or minus electricity of the outside. This theory of the Leyden phial Franklin established in the clearest manner, by showing that the outside and the inside coating possessed opposite electricities, and that, in charging it, exactly as much electricity is added on one side as is subtracted from the other. The abundant discharge of electricity by points was observed by Franklin in his earliest experiments, and also the power of points to conduct it copiously from an electrified body. Hence he was furnished with a simple method of collecting electricity from other bodies; and he was thus enabled to perform those remarkable experiments which we shall now proceed to explain. Hawksbee, Wall, and Nollet had successively suggested the similarity between lightning and the electric spark, and between the artificial snap and the natural thunder. Previous to the year 1750 Franklin drew up a statement, in which he showed that all the general phenomena and effects which were produced by electricity had their counterpart in lightning. After waiting some time for the erection of a spire at Philadelphia, by means of which he thought to bring down the electricity of a thunder-storm, he conceived the idea of sending up a kite among the clouds themselves. With this view he made a small cross of two small light strips of cedar, the arms being sufficiently long to reach to the four corners of a large thin silk handkerchief when extended. The corners of the handkerchief were tied to the extremities of the cross, and when the body of the kite was thus formed, a tail, loop, and string were added to it. The body was made of silk to enable it to bear the violence and wet of a thunder-storm. A very sharp pointed wire was fixed at the top of the upright stick of the cross, so as to rise a foot or more above the wood. A silk ribbon was tied to the end of the twine next the hand, and a key suspended at the junction of the twine and silk. In company with his son, Franklin raised the kite like a common one, in the first thunder-storm, which happened in the month of June 1752. To keep the silk ribbon dry, he stood within a door, taking care that the twine did not touch the frame of the door; and when the thunder-clouds came over the kite he watched the state of the string. A cloud passed without any electrical indications, and he began to despair of success. He saw, however, the loose filaments of the twine standing out every way, and he found them to be attracted by the approach of his finger. The suspended key gave a spark on the application of his knuckle, and when the string had become wet with the rain, the electricity became abundant; a Leyden jar was charged at the key, and by the electric fire thus obtained spirits were inflamed, and all the other electrical experiments performed which had been formerly made by excited electrics. In subsequent trials with another apparatus, he found that the clouds were sometimes positively and sometimes negatively electrified, and so demonstrated the perfect identity of lightning and electricity. Having thus succeeded in drawing the electric fire from the clouds, Franklin conceived the idea of protecting buildings from lightning by erecting on their highest parts pointed iron wire or conductors communicating with the ground. The electricity of a hovering or a passing cloud would thus be carried off slowly and silently; and if the cloud was highly charged, the lightning would strike in preference the elevated conductors.

The most important of Franklin’s electrical writings are his Experiments and Observations on Electricity made at Philadelphia, 1751-54; his Letters on Electricity, and various memoirsand letters, Phil. Trans., 1756, 1760, Ac.

About the same time that Franklin was making his kite experiment in America, D’Alibard and others in France had erected a long iron rod at Marli, and obtained results agreeing with those of Franklin. Similar investigations were pursued by many others, among whom Father Beccaria deserves especial mention.

These experiments were often dangerous, and in one case ' a fatal accident occurred. Professor Richman of St Petersburg had erected on his house an iron rod to collect the 1 electricity of thunder-clouds. On the 6th August 1753, during a thunder-storm, he was observing, along with his friend Sokolow, the indications of an electrometer which formed part of his apparatus, when a tremendous thunder clap burst over the neighbourhood. Richman bent to observe the electrometer; while in this position, his head being a foot from the iron rod, Sokolow saw a globe ol bluish fire about the size of the fist shoot from the iron rod to the professor’s head, with a report like that of a pistol, The shock was fatal; Richman fell back upon a chest and instantly expired. Sokolow was stupified and benumbed, and the red hot fragments of a metallic wire struck his clothes, and covered them with burnt marks.

One of the most diligent labourers in the field of electrical science was an Englishman, John Canton {Phil. Trans., 1753-54). Before his time it had been assumed as indisputable that the same kind of electricity was invariably produced by the friction of the same electric,—that glass, for example, yielded always vitreous, and amber always resinous electricity. Having roughened a glass tube by grinding its surface with emery and sheet lead, he found that it possessed vitreous or positive electricity when excited with oiled silk, but resinous electricity when excited with new flannel. He found, in short, that vitreous or resinous electricity might, in certain cases, be developed at will in the same tube, by altering the surfaces of the tube and the exciting rubber. Removing the polish from one half of the tube, he excited the different electricities with the same rubber at a single stroke, and, curiously enough, the rubber was found to move much more easily over the rough than over the polished half. Canton likewise discovered that glass, amber, sealing-wax, and calcareous spar were all electrified positively when taken out of mercury; and hence he was led to the important practical discovery that an amalgam of mercury and tin was most efficacious in exciting glass when applied to the surface of the rubber. Canton discovered, and to a certain extent explained by the then prevalent theory of “electrical atmospheres,” the fundamental fact of electrification by induction. He also found that the air in a room could be electrified positively or negatively, and might remain thus electrified for a considerable time.

Beccaria, a celebrated Italian physicist, kept up the spirit of electrical discovery in Italy. He showed that water is a very imperfect conductor of electricity, that its conducting power is proportional to its quantity, and that a small quantity of water opposes a powerful resistance to the passage of electricity. He succeeded in making the electric spark visible in water, by discharging shocks through wires that nearly met in tubes filled with water. In this experiment the tubes, though sometimes eight or ten lines thick, were burst in pieces. Beccaria likewise demonstrated that air adjacent to an electrified body gradually acquired the same electricity, that the electricity of the body is diminished by that of the air, and that the air parts with its electricity very slowly. He considered that there was a mutual repulsion between the particles of the electric fluid and those of air, and that in the passage of the former through the latter a temporary vacuum was formed. Beccaria’s experiments on atmospherical electricity are of the greatest interest to the meteorologist. For farther account of his work, see his Lettere deW Elettr., 1758; Experimenta, 1772; and letters, <kc., in Phd. Trans, about 1770.

The science of electricity owes several practical as well as theoretical observations to Robert Symmer {Phil. Trans., about 1759). In pulling off his stockings in the evening, he had often remarked that they not only gave a crackling noise, but even emitted sparks in the dark. The electricity was most powerful when a silk and a worsted stocking had been worn on the same leg, and it was best exhibited by putting the hand between the leg and the stockings, and pulling them off together. The one stock ing being then drawn out of the other, they appeared more or less inflated, and exhibited the attractions and repulsions of electrified bodies. Two white silk stockings, or two black ones, when put on the same leg and taken off, gave no electrical indications. When a black and a white stocking were put on the same leg, and after ten minutes taken off', they were so much inflated when pulled asunder, that each showed the entire shape of the leg, and at the distance of a foot and a half they rushed to meet each other.

“But what appears most extraordinary is, that when they are separated, and removed at a certain distance from each other, their electricity does not appear to have been in the least impaired by the shock they had in meeting. They are again inflated, again attract and repel, and are as ready to rush together as before. When this experiment is performed with two black stockings in one hand, and two white in the other, it exhibits a very curious spectacle; the repulsion of those of the same colour, and the attraction of those of different colours, throws them into an agitation that is not unentertaining, and makes them catch each at that of its opposite colour, at a greater distance than one would expect. When allowed to come together, they all unite in one mass. When separated, they resume their former appearance, and admit of the repetition of the experiment as often as you please, till their electricity, gradually wasting, stands in need of being recruited.

Symmer likewise found that a Leyden jar could be charged by the stockings either positively or negatively, according as the wire from the neck of the jar was presented to the black or the white stocking. When the electricity of the white stocking was thrown into the jar, and then the electricity of the black one, or vice versa, the jar was not electrified at all. With the electricity of two stockings he charged the jar to such a degree that the shock from it reached both his elbows; and by means of the electricity of four silk stockings he kindled spirits of wine in a tea-spoon which he held in his hand, and the shock was at the same time felt from the elbows to the breast. Symmer has the merit of having first maintained the theory of two distinct fluids, not independent of each other, as Dufay supposed them to be, but co-existent, and, by counteracting each other, producing all the phenomena of electricity. He conceived that when a body is said to be positively electrified, it is not simply that it is possessed of a larger share of electric matter than in a natural state, nor, when it is said to be negatively electrified, of a less; but that, in the former case, it is possessed of a larger portion of one kind of electricity, and in the latter, of a larger portion of the other; while a body, in its natural state, remains unelectrified, because there is an equal amount of the two everywhere within it.

Contemporary with Symmer were Délavai, Wilson, Cigna, Kinnersley, Wilcke, and Priestley (for the works of these electricians consult Young). Délavai found that the sides of vessels that were perfect conductors were nonconductors, and that animal and vegetable bodies lost their conducting power when reduced to ashes. Wilson concluded that when two electrics are rubbed together, the harder of the two is generally electrified positively and the other negatively, the electricities always being opposite. Cigna made many curious experiments by using silk ribbands in place of the silk stockings of Symmer. Kinnersley, the friend of Franklin, made some important experiments on the elongation and fusion of iron wires, when a strong charge was passed through them in a state of tension {Phil. Trans., 1763); he also experimented on the disruptive discharge in air. Wilcke brought to light many phenomena respecting the electrification produced by the melting of electric substances.

The pyro-electricity of minerals, or the faculty possessed by some minerals of becoming electric by heat, and of exhibiting negative and positive poles, now began to attract ‘ the notice of philosophers. There is reason to believe that the lyncumum of the ancients, which, according to Theophrastus, attracted light bodies, was the tourmaline, a Ceylon mineral, in which the Dutch had early recognized the same attractive property, whence it got the name of Aschentriklcer, or attractor of ashes. In 1717 Μ. Lemery exhibited to the Academy of Sciences a stone from Ceylon which attracted light bodies; and Linnaeus, in mentioning the experiments of Lemery, gives the stone the name of Lapis Electricus. The Duke de Noya was led in 1758 to purchase some of the stones called tourmaline in Holland, and, assisted by Daubenton and Adanson, he made a series of experiments with them, a description of which was published. The subject, however, had engaged the attention of Aepinus, a celebrated German philosopher, who published an account of them in 1756. Hitherto nothing had been said respecting the necessity of heat to excite the tourmaline; but it was shown by Aepinus that a temperature between 99⅛ o and 212° Fahr, was requisite for the development of its attractive powers. Benjamin Wilson {PhU. Trans., 1763, l fcc.), Priestley, and Canton continued the investigation; but it was reserved for the Abbé Haiiy to throw a clear light on this curious branch of the science {Trade de Mineralogie). He found that the electricity of the tourmaline decreased rapidly from the summits or poles towards the middle of the crystal, where it was imperceptible; and he discovered that if a tourmaline is broken into any number of fragments, each fragment, when excited, has two opposite poles. Haiiy discovered the same property in the Siberian and Brazilian topaz, borate of magnesia, mesotype, prehnite, sphene, and calamine. He also found that the polarity which minerals receive from heat has a relation to the secondary forms of their crystals,—the tourmaline, for example, having its resinous pole at the summit of the crystal which has three faces, and its vitreous pole at the summit which has six faces. In the other pyroelectrical crystals above mentioned, Haiiy detected the same deviation from the rules of symmetry in their secondary crystals which occurs in tourmaline. Brard discovered that pyro-electricity was a property of the axinite; and it was afterwards detected in other minerals. In repeating and extending the experiments of Haiiy, Sir David Brewster discovered that various artificial salts were pyro-electrical; and he mentions tartrate of potash and soda, and tartaric acid, as exhibiting this property in a very strong degree. He also made many experiments with the tourmaline when cut into thin slices, and reduced to the finest powder, in which state each particle preserved its pyro-electricity; and he showed that scolezite and mesolite, even when deprived of their water of crystallization and reduced to powder, preserve their property of becoming electrical by heat. When this white powder is heated and stirred about by any substance whatever, it collects in masses like new fallen snow, and adheres to the body with which it is stirred. (For Sir David Brewster’s work on pyro-electricity see Trans. RH.E., 1845; Phil. Mag., Dec. 1847; Edinburgh Journal of Science, Oct. 1824 and 1825).

In addition to his experiments on the tourmaline, Aepinus made several on the electricity of melted sulphur; and in conjunction with Wilcke, he investigated the subject of electric atmospheres, and discovered a beautiful method of charging a plate of air by suspending large wooden boards coated with tin, and having their surfaces near each other and parallel. Aepinus, however, has been principally distinguished by his ingenious theory of electricity, which he has explained and illustrated in a separate work {Tentamen Theorioe Electricitatis et Magnetismi) which appeared at St Petersburg in 1759. This theory is founded on the following principles. 1. The particles of the electric fluid repel each other with a force decreasing as the distance increases. 2. The particles of the electric fluid attract the particles of all bodies, and are attracted by them, with a force obeying the same law. 3. The electric fluid exists in the pores of bodies; and while it moves without any obstruction in non-electrics. such as metals, water, &c., it moves with extreme difficulty in electrics, such as glass, rosin, ιfcc. 4. Electrical phenomena are produced either by the transference of the fluid from a body containing more to one containing less of it, or from its attraction and repulsion when no transference takes place.

The electricity of fishes, like that of minerals, now began to excite very general attention. The ancients, as we have seen, were acquainted with the benumbing power of the torpedo, but it was not till-1676 that modern naturalists attended to this remarkable property. The Arabians had long before given this fish the name of raad or lightning; but Redi was the first who communicated the fact that the shock was conveyed to the fisherman by means of the line and rod which connected him with the fish. Lorenzini published engravings of its electrical organs; Reaumur described the electrical properties· of the fish; Kämpfer compared the effects which it produced to lightning; but Bancroft was the first person who distinctly suspected that the effects of the torpedo were electrical. In 1773 Walsh {Phil. Trans., 1773-5) and Ingenhousz proved, by many curious experiments, that the shock of the torpedo was an electrical one; and Hunter {PhU. Trans., 1773-5) examined and described the anatomical structure of its electrical organs. Humboldt {Ann. de Chim. et de Phys., i. 15), Gay-Lussac, and Geoffroy pursued the subject with success; and Cavendish {Phil. Trans., 1776) constructed an artificial torpedo, by which he imitated the actions of the living animal. The subject was also investigated by Todd, Sir Humphrey Davy {Phil. Trans., 1829), John Davy, and Faraday {Exp. Res., vol. ii.). The power of giving electric shocks has been discovered also in the Gymηotus electricus, the Malapterurus electricus, 2 the Trichiurus electricus, and the Tetraodon electricus? The most interesting and the best known of these singular fishes is the Gyrnnol·us or Surinam eel. Humboldt gives a very graphic account of the combats which are carried on in South America between the gymnoti and tbe wild horses in the vicinity of Calabozo.

Among the cultivators of electricity Henry Cavendish is entitled to a distinguished place. Before he had any knowledge of the theory of Aepinus, he had communicated to the Royal Society a similar theory of electrical phenomena. As, however, he had carried the theory much further, and considered it under a more accurate point of view, he did not hesitate to give his paper to the world {Phil. Trans., 1771). Cavendish made some accurate experiments on the relative conducting power of different substances. He found that electricity experiences as much resistance in passing through a column of water one inch long as it does in passing through an iron wire of the same diameter 400,000,000 inches long, whence he concluded that iron wire conducts 400,000,000 times as well as rain or distilled water. He found that a solution of one part of salt in one of water conducts a hundred times better than fresh water, and that a saturated solution of sea-salt conducts seven hundred and twenty times better than fresh water. Cavendish likewise determined by nice experiments that the quantity of electricity on coated glass of a certain area increased with the thinness of the glass, and that on different coated plates the quantity was as the area of the coated surface directly, and as the thickness of the glass inversely. , Although electricity Lad been employed as a chemical agent in the oxidation and fusion of metals, yet it is to Cavendish that we owe the first of those brilliant inquiries which have done so much for the advancement of modem chemistry. By using different proportions of oxygen and hydrogen, and examining the products which they formed after explosion with the electric spark, he obtained a proportion of which the product was pure water (P7u7. Trans., 1784-5). The decomposition of water by the electric spark was first effected by Paets Van Troostwijk and Deiman; improved methods of effecting it were discovered and used by Pearson, Cuthbertson, and Wollaston (Phil. Trans., 1801).

The great discovery made by Galvani in 1790, that the contact of metals produced muscular contraction in the frog, and the invention of the voltaic pile, in 1800, by Volta led to the recognition of a new kind of electricity called Galvanic or Voltaic Electricity, which is now proved to be identical with frictional electricity. The chemical effects of the voltaic pile far transcend those of ordinary electricity. In 1800 Nicolson and Carlisle discovered the power of the pile to decompose water; and in 1807 (Bαkcrian Lecture) Sir Humphry Davy decomposed the earths and the alkalies, and thus created a new epoch in the history of chemistry.

J Contemporaneous with Cavendish was Coulomb, one of the most eminent experimental philosophers of the last century. In order to determine the law of electrical action, he invented an instrument called a torsion balance, which has since his time been universally used in all delicate researches, and which is particularly applicable to the measurement of electrical and magnetical actions. Aepinus and Cavendish had considered the action of electricity as diminishing with the distance; but Coulomb proved, by a series of elaborate experiments, that it varied, like gravity, in the inverse ratio of the square of the distance. Dr Robison had previously determined, without, however, having published his experiments, that in the mutual repulsion of two similarly electrified spheres, the law was slightly in excess of the inverse duplicate ratio of the distance, while in the attraction of oppositely electrified spheres the deviation from that ratio wa≡ in defect; and hence he concluded that the law of electrical action was similar to that of gravity. Adopting the hypothesis of two fluids, Coulomb investigated experimentally and theoretically the distribution of electricity on the surface of bodies. He determined the law of its distribution between two conducting bodies in contact; he measured the density of the electricity at different points of two spheres in contact; he ascertained the distribution of electricity among several spheres (whether equal or unequal) placed in contact in a straight line; he measured the distribution of electricity on the surface of a cylinder, and its distribution between a sphere and cylinder of different lengths but of the same diameter. His experiments on the dissipation of electricity possess also a high value. He found that the momentary dissipation was proportional to the degree of electrification at the time, and that, when the charge was moderate, its dissipation was not altered in bodies of different kinds or shapes. The temperature and pressure of the atmosphere did not produce any sensible change; but he concluded that the dissipation was nearly proportional to the cube of the quantity of moisture in the air. In examining the dissipation which takes place along imperfectly insulating substances, he found that a thread of gum-lac was the most perfect of all insulators; that it insulated ten times as well as a dry silk thread; and that a silk thread covered with fine sealing-wax insulated as powerfully as gum-lac when it had four times its length. He found also that the dissipation of electricity along insulators was chiefly owing to adhering moisture, but in some measure also to a slight conducting power. For the memoirs of Coulomb see Mém. de Math, et Phys, de l ’ Acad. de Sc., 1785, &c.

Towards the end of the last century a series of experiments was made by Laplace, Lavoisier, and Volta (Phil. Trans., 1782, or Collezione dell , Op.), from which it appeared that electricity is developed when solid or fluids bodies pass into the gaseous state. The bodies which were to be evaporated or dissolved were placed upon an insulating stand, and made to communicate by a chain or wire with a Cavallo’s electrometer, or with Volta’s condenser, when it was suspected that the electricity increased gradually. When sulphuric acid diluted with three parts of water was poured upon iron filings, hydrogen was disengaged with a brisk effervescence; and at the end of a few minutes the condenser was so highly charged as to yield a strong spark of negative electricity Similar results were obtained when charcoal was burnt on a chafing dish. Volta, who happened to be at Paris when these experiments were made, and who took an active part in them, subsequently observed that· the electricity produced by evaporation was always negative. He found that burning charcoal gives out negative electricity; and in other kinds of combustion he obtained distinct electrical indications. In this state of the subject Saussure (Voyage dans les Alpes, t. ii. p. 808, et seqqj) undertook a series of elaborate experiments on the electricity of evaporation and combustion. In his first trials he found that the electricity was sometimes positive and sometimes negative when water was evaporated from a heated .crucible of iron; but he afterwards found it to be always positive both in an iron and a copper crucible. In a silver and a porcelain crucible the electricity was negative. The evaporation of alcohol and of ether in a silver crucible also gave negative electricity. Saussure made many fruitless trials to obtain electricity from combustion, and he likewise failed in his attempt to procure it from evaporation without ebullition. Many valuable additions were about this time made to electrical apparatus, as well as to the science itself, by Van Marum, Cavallo, Nicholson, Cuthbertson, Brooke, Bennet, Read, Morgan, Henley, and Lane; but these cannot here be noticed in detail.

The application of analysis to electrical phenomena may be dated from the commencement of the present century. Coulomb had considered only the distribution of electricity on the surface of spheres; but Laplace undertook to investigate its distribution on the surface of ellipsoids of revolution, and he showed that the thickness of the coating of fluid at the pole was to its thickness at the equator as the polar is to the equatorial diameter. Biot (Traité de Physique Exp. et Math.) has extended this investigation to all spheroids differing little from a sphere, whatever may be the irregularity of their figure. He likewise determined analytically that the losses of electricity form a geometrical progression when the two surfaces of ajar or plate of coated glass are discharged by successive contacts; and he found that the same law regulates the discharge when a series of jars or plates are placed in communication with each other. It is to Poisson (Mem. de l , Iηst. Math, et Phys., 12, 1811, <fcc.) however, that we are mainly indebted for having brought the phenomena of electricity under the dominion of analysis, and placed it on the same level as the more exact sciences. Assuming the hypothesis of two fluids, he deduced theorems for determining the distribution of the electric fluid on the surface of two conducting spheres when they are either placed in contact or at any given distance. The truth of these theorems had been established by experiments performed by Coulomb long before the theorems themselves had been investigated.

Voltaic electricity had now absorbed the attention of experimental philosophers. The splendour of its phenomena, as well as its association with chemical discovery, contributed to give it popularity and importance; but the discoveries of Galvani and Volta were destined, in their turn, to pass into the shade, and the intellectual enterprise of the natural philosophers of Europe was directed to new branches of electrical and magnetical science. Guided by; theoretical anticipations, Professor H. C. Oersted of Copen-lιagen {Experimenta circa effectum confictus electrici in acum magneticam) in 1820 discovered that the electrical current of a galvanic battery, when made to pass through a platinum wire, acted upon a compass needle placed below the wire. He found that a magnetic needle placed in the neighbourhood of an electric current always places itself perpendicular to the plane through the current and the centre of the needle; or, more definitely, that a magnetic north pole, carried at a constant distance round the current in the direction of rotation of an ordinary cork-screw advancing in the positive direction of the current, would always tend to move in the direction in which it is being carried.

Scarcely had the news of Oersted’s discovery reached France when a French philosopher, Ampère, set to work to develop the important consequences which it involved. Physicists had long been looking for the connection between magnetism and electricity, and had, perhaps, inclined to the view that electricity was somehow to be explained as a magnetic phenomenon. It was, in fact, under the influence of such ideas that Oersted was led to his discovery. Ampère showed that the explanation was to be found in an opposite direction. He discovered the ponderomotive action of one electric current on another, and by a series of well-chosen experiments he established the elementary laws ef electrodynamical action, starting from which, by a brilliant train of mathematical analysis, he not only evolved the complete explanation of all the electromagnetic phenomena observed before him, but predicted many hitherto unknown. The results of his researches may be summarized in the statement that an electric current in a linear circuit of any form is equivalent in its action, whether on magnets or other circuits, to a magnetic shell bounded by the circuit, whose strength at every point is constant and proportional to the strength of the current. By his beautiful theory of molecular currents, he gave a theoretical explanation of that connection between electricity and maguetism which had been the dream of previous investigators. If we except the discovery of the laws of the induction of electric currents made about ten years later by Faraday, no advance in the science of electricity can compare for completeness and brilliancy with the work of Ampère. Our admiration is equally great whether we contemplate the clearness and power of his mathematical investigations, the aptness and skill of his experiments, or the wonderful rapidity with which he elaborated his discovery when he had once found the clue.

In 1821 Faraday, who was destined a little later to do so much for the science of electricity, discovered electromagnetic rotation {Quarterly Journal, xii.), having succeeded in causing a horizontal wire carrying a current to rotate continuously across the vertical lines of a field of magnetic force. The experiment was very soon repeated in a variety of forms by De la Rive, Barlow, Ritchie, Sturgeon, and others; and Davy {Phil. Trans.), in 1823, observed that, when two wires connected with the pole of a battery were dipped into a cup of mercury placed on the pole of a powerful magnet, the fluid metal rotated in opposite directions about the two electrodes. The rotation of a magnet about a fixed current and about its own axis was at once looked for, and observed by Faraday and others. The deflection of the voltaic arc by the magnet had been observed by Davy in 1821 {Phil. Trans.)·, and in 1840 Walker observed the rotation of the luminous discharge in a vacuum tube. For many beautiful experiments on the influence of the magnet on the strata, <tc., in vacuum tubes, we are indebted to Pliicker, De la Rive, Grove, Gassiot, and others who followed them.

One of the first machines in which a continuous motion was produced by means of the repulsions and attractions between electromagnets and fixed magnets or electromagnets was invented by Ritchie {Phil. Trans., 1833). The artifice in such machines consists in reversing the polarity of one of the electromagnets when the machine is near the position of equilibrium. For a general theory of these machines, showing the reasons why they are not useful as economic motive powers, see Jacobi {Mémoire sur l'Apρlicatiοn de l' Électro-magnétisme au Mouvement des Machines, Potsdam, 1835), and Joule {Mech. Mag., xxxvi.). Electro-magnetic engines have, however, found a restricted use in scientific workshops, such as Froment’s, in driving telegraphic apparatus, &c.

In 1820 Arago {Ann. de Chim. et de Phys., t. xv.) and Davy {Annals of Philosophy, 1821) discovered independently the power of the electric current to magnetize iron and steel. Savary {Aim. de Chim. et de Phys., t. xxxiv., 1827) made some very curious experiments on the alternate directions of magnetization of needles placed at different distances from a wire conveying the discharge of a Leyden jar. The dependence of the intensity of magnetization on the strength of the current was investigated by Lenz and Jacobi {Pogg. Ann., xlvii., 1839), and Joule found that magnetization did not increase proportionately with the current, but reached a maximum (Sturgeon’s Aim. of El. iv. 1839). The farther development of this subject, which really belongs to magnetism, has been carried on by Weber, Müller, Von Waltenhofen, Dub, Wiedemann, Quintus Icilius, Riecke, Stoletow, Rowland, and others. The use of a core of soft iron, magnetized by a helix surrounding it, has become universal in all kinds of electrical apparatus. Electromagnets of great power have in this way been constructed and used in electrical researches by Brewster, Sturgeon, Henry, Faraday, and others.

The most illustrious among the successors of Ampère was Wilhelm Weber. He greatly improved the construction of the galvanometer, and invented the electro-dynamometer. To these instruments he applied the mirror scale and telescope method of reading, which had been suggested by Poggendorff, and used by himself and Gauss in magnetic measurements about 1833. In 1846 he proceeded with his improved apparatus to test the fundamental laws of Ampère. The result of his researches was to establish the truth of Ampère’s principles, as far as experiments with closed circuits could do so, with a degree of accuracy far beyond anything attainable with the simple apparatus of the original discoverer. The experiments of Weber must be looked upon as the true experimental evidence for the theory of Ampère, and as such they form one of the corner-stones of electrical science.

While experiment was thus busy, theory was not idle. In 1845 Grassmann published {Pogg. Aim., lxiv.) his Neue Theorie der Electrodynamik, in which he gives an elemen-1 tary law different from that of Ampère, but leading to the 1 same results for closed circuits. In the same year F. E. Neumann published yet another law. In 1846 Weber announced his famous hypothesis connecting electrostatical and electrodynamical phenomena. Much has been written on the subject by Carl Neumann, Riemann, Stefan, Clausius, and others. Very important are three memoirs by Helmholtz, in Crellds Journal (1870-2-4), in which a general view is taken of the whole question, and the works of his predecessors are critically handled. We shall have occasion, in the body of the article, to refer to the dynamical theory of Clerk Maxwell, which promises to effect a revolution in this part of electrical science.

By his discovery of thermo-electricity in 1822 (Pogg. Aim., vi.), Seebeck opened up a new department He found that when two different metals are joined in circuit there will be an electric current in the circuit if the junctions are not at the same temperature; he arranged the metals in a thermo-electric series, just as Volta and his followers had arranged them in a contact series. Cumming (Annals of Phil., 1823) found that the order of the metals was not the same at different temperatures. This phenomenon has been called thermo electric inversion. In 1834 Peltier discovered that if a current be sent round a circuit of two metals in the direction in which the thermo-electromotive force would naturally send it, then the hot junction is cooled, and the cool junction heated. This effect, which is reversible, and varies as the strength of the current, is called the Peltier effect. Sir W. Thomson made many experiments on thermo-electricity, and applied to the experimental results the laws of the dynamical theory of heat. His reasonings led him to predict a new thermo-electric phenomenon, the actual existence of which he afterwards verified by an elaborate, series of very beautiful experiments (Phil. Trans., 1856). He has given a general theory of the thermo-electric properties of matter, taking into account the effect of structure, Ac. His experimental researches have been ably continued by 1’rofessor Tait, who, guided by theoretical considerations to the conjecture that the curves in what Thomson called the “thermoelectric diagram” must be straight lines, made an extended series of experiments, and showed that they were in general very approximately either straight lines or made up of pieces of straight lines. Our knowledge of thermo-electricity has been advanced by Becquerel, Magnus, Matthiessen, Leroux, Avenarius, and others. Thermo-electric batteries of considerable power have been constructed by Markus, Noë, and Clamond, and employed more or less in the arts.

In 1824 Arago (Ann. de Chim. et de Phys., t. xxvii. Ac.) made a remarkable discovery, which led ultimately to results of the greatest importance. He found that when a magnetic needle is suspended over a rotating copper disc the needle tends to follow the motion of the disc. This phenomenon, which has been called the “magnetism of rotation,” excited great interest; Barlow (Phil. Trans., 1825), Herschel, Seebeck (Pogg. Ann., vii., 1826), and Babbage (Phil. Trans., 1825) made elaborate researches on the subject; and Poisson (Mém. de l'Acad., vii., 1826) attempted to give a theoretical explanation in his memoir on magnetism in motion. The true explanation was not arrived at until Faraday took up the subject a little later. We may mention, here, however, the experiments of Plücker, Matteucci, and Foucault on the damping of the motions of masses of metal between the poles of electromagnets. The damping of a compass needle suspended over a copper plate, observed by Seebeck (l.c.), has been taken advantage of in the construction of galvanometers.

In 1831 Faraday began, with the discovery of the induction of electric currents, that brilliant series of experimental researches which has rendered his name immortal. The first experiment which he describes was made with two helices of copper wire wound side by side on a block of wood, and insulated from each other by intervening layers of twine. One of these helices was connected with a galvanometer, and the other with a battery of a hundred plates, and it was found that on making and breaking the battery circuit a slight sudden current passed through the galvanometer in opposite directions in the two cases. He also discovered that the mere approach or removal of a circuit carrying a current would induce a current in a neighbouring closed circuit, and that the motion of magnets produces similar effects. To express in a concise manner his discoveries, Faraday invented his famous conception of the lines of magnetic force, or lines the direction of which at any point of their course coincides with that of the magnetic force at that point. His discovery can be thus stated :—Whenever the number of lines of force passing through a closed circuit is altered, there is an electromotive force tending to drive a current through the circuit, whose direction is such that it would itself produce lines of force passing through the circuit in the opposite direction. Nothing in the whole history of science is more remarkable than the unerring sagacity which enabled Faraday to disentangle, by purely experimental means, the laws of such a complicated phenomenon as the induction of electric currents. The wonder is only increased when we look to his papers, and find the first dated November 1831, and another January 1832, in which he shows that he is in complete possession of all the general principles that are yet known on the subject. Faraday very soon was able to show that the current developed by induction had all the properties of the voltaic current, and he made an elaborate comparison of all the different kinds of electricity known,—statical, dynamical or voltaic, magneto-, thermo-, and animal electricity,—showing that they were identical so far as experiment could show. In 1833 Lenz made a series of important researches (Pogg. Ann., xxxi., 1834, xxxiv., 1835), which, among other results, led him to his celebrated law by means of which the direction of the induced current can be predicted from the theory of Ampère, the rule being that the direction of the induced current is always such that its electromagnetic action tends to oppose the motion which produces it. This law leads to the same results as the principles of Faraday. The researches of Ritchie and Henry about this time, and of Dove a little later, are also of importance. In 1845 F. E. Neumann did for magnetoelectric induction what Ampère did for electrodynamics, by developing from the experimental laws of Lenz the mathematical theory of the subject (Abh. der Berl. Ahad, der Wissenschaft, 1845-7). He discovered a function which has been called the “potential” (of one linear current on another or on itself), from which he deduced a theory of induction completely in accordance with experiment. About the same time Weber deduced the mathematical laws of ihduction from his elementary law of electrical action, which, as we have already seen, he applied to explain electrostatic and electromagnetic action. In 1846 Weber, applying his improved instruments, arrived at accurate verifications of the laws of induction, which by this time had been developed mathematically by Neumann and himself. In 1849 Kirchhoff determined experimentally in a certain case the absolute value of the current induced by one circuit in another; and in the same year Edlund made a series of careful experiments on the currents of self and mutual induction, which led to the firmer establishment of the received theories. Helmholtz gave the mathematical theory of the course of induced currents in various cases, and made a series of valuable experiments in verification of his theory (Pogg. Ann., lxxxiii., 1851). Worthy of mention here are also the experiments and reasonings of Felici in 1852. In the Philosophical Magazine for 1855, Sir W. Thomson investigated mathematically the discharge of a Leyden jar through a linear conductor, and predicted that under certain circumstances the discharge would consist of a series of decaying oscillations. This oscillatory discharge was observed in 1857 by Feddersen (Pogg. Ann., cviii.) The law of Weber has been applied by Kirchhoff to the case of conductors in three dimensions. The most important of all the recent contributions to this part of electrical science is the theory of Clerk Maxwell, which aims at deducing the phenomena of the electromagnetic field from purely dynamical principles with the aid of the fewest possible hypotheses {Phil. Trans., 1864; Electricity and Magnetism, 1873). He has established the general equations which determine the state of the electric field, and he has by means of these equations constructed an electromagnetic theory of light, which is full of suggestions for the philosopher, whether speculative or experimental. The theory of Helmholtz, and his valuable criticisms on the works of those that have laboured in this department, are to be found in three memoirs already alluded to.

Magneto-electricity has been largely applied in the arts. One of the first machines for producing electricity by induction was made by Pixii. It consisted of a fixed horseshoe armature wound with copper wire, in front of which revolved about a vertical axis a horseshoe magnet. The machine was furnished with a commutator for delivering the alternating currente in a common direction. By means of this machine Faraday and Hachette decomposed water and collected the disengaged gases separately. Many variations of this type of machine were constructed by Ritchie, Saxton, Clark, Von Ettingshausen, Stöhrer, Dove, Wheatstone, and others. In 1857 Siemens effected a great improvement by inventing the form of armature which bears his name. The next improvement was to replace the fixed magnets by electromagnets, the current for which was furnished by a small auxiliary machine. Wilde’s machine (1867) is of this kind. Siemens, Wheatstone, and others suggested that the fixed electromagnet should be fed by a coil placed on the armature itself, so that starting from the residual magnetism of the armature the machine goes on increasing its action up to a certain point. Ladd’s machine (1867) is constructed on this principle. The most recent of these machines is that of Gramme, the peculiarity of which is that the coil of the armature is divided up into a series of coils arranged round an axis, the object being to produce a continuous instead of a fluctuating current. It has been proposed of late to employ electromagnetic machines in lighting streets and workshops, and the experiment has been tried with some success. They have been employed for some time back in lighthouse work. The most important inductive apparatus for the physicist is the induction coil or inductorium, which has been brought to great perfection in the workshop of Ruhmkorff. Poggendorff {Annalen, 1855) suggested several improvements in this kind of apparatus. Fizeau, who added the condenser (1853), Foucault, who designed the interrupter which bears his name (1855), and Ritchie, who devised the plan of dividing the coil into sections by insulating partitions, have all aided in bringing the instrument to perfection. Very powerful machines of this kind have been constructed. Λ large one in the Polytechnic Institution, London, gives a 29-inch spark, and one recently constructed by Apps for Mr Spottiswoode gives a spark of 42 inches. The mathematical theory of magneto-electric machines has been treated by Maxwell {Proc. Poy. S 0 c., 1867). He has also given a theory of the action of the condenser in the inductorium {Phil. Mag., 1868). Two papers by Strutt (now Lord Rayleigh) in Phil. Mag., 1869—70, are very interesting in connection with the same subject.

In the year 1827 Dr G. S. Ohm rendered a great service to the science of electricity by publishing his mathematical theory of the galvanic circuit {Die Galvanische Kette mathematisch bearbeitet). Before his time the quantitative circumstances of the electric current had been indicated in a very vague way by the use of the terms “intensity” and “quantity,” to which no accurately defined meaning was attached. Ohm’s service consisted in introducing and defining the accurate notions—electromotive force, current strength, and resistance. He indicated the connection of these with experiment, and stated his famous law that the electromotive force divided by the resistance is equal to the strength of the current. The theory on which. Ohm based his law may be and has been disputed, but the law itself and the applications which Ohm and others have made of it are in the fullest agreement with all known facts. The merit of Ohm really consists in having satisfactorily analysed a great group of phenomena which had up to his time baffled all those who attempted the task. How great his service was is easily seen when we remark the progress of those who adopted his ideas as compared with those who for a time hesitated to do so. Ohm was guided in his mathematical work by analogy with the problem of the flux of heat, and introduced for the first time into the theory of the pile, the equivalent of the modern -word potential. Ohm’s word was electroscopicforce or tøm'on {Spannung), and he showed that the fall of the potential is uniform along a homogeneous linear conductor. He considered that the potential was analogous to the temperature, and the flow of electricity to the flow of heat, so that the former just as much as the latter obeys the law of continuity. Ohm verified his theoretical conclusions with thermo-electric piles, and he observed, as Erman {Gilb. Ann., 1801) had done before him, the differences of potential at different points of the circuit. Davy, Pouillet, and Becquerel laboured at the experimental verification of Ohm’s law, and a great body of evidence was given by Fechner in his Maasbestimmungen über die Galvanische Kette (1831). The law of the fall of potential was verified by the elder Kohlrausch, who employed in his researches Volta’s condenser and Dellmann’s electrometer {Pogg. Ann., lxxv., 1848). Later researches of a similar nature were made by Gaugain and Branly. Among recent investigations bearing on Ohm’s law, the most remarkable is the verification for electrolytes by Kohlrausch (the younger) and Nippoldt. They principally used alternating currents in their researches, which were furnished by a “sine inductor,” the measuring instrument employed being the electro-dynamometer of Weber. In the report of the British Association for 1876 an account is given of some experiments, in which the testing of this law seems to have been carried to the limit of experimental resources. It must now be allowed to rank with the law of gravitation and the elementary laws of statical electricity as a law of nature in the strictest sense. Many remarkable applications of Ohm’s law have been made of late, in particular to linear conductors by Ohm, Poggendorff, and especially Kirchhoff {Pogg. Aim., 1845-7-8). The works of Helmholtz, Smaasen, and Kirchhoff on conduction in three dimensions must also be mentioned. Very important, on account of the experimental results with which they deal, are the calculations of Du Bois Reymond {Pogg., lxxi., 1845) and Riemann {Werke, Leipsic, 1876) on Nobili’s rings, and of Kirchhoff {Pogg., lxvii., 1848), W. R. Smith {Proc. Poy. Soc. Edm., 1869-70), Quincke, Stefan, Adams, and others on conduction in plates. Theoretical applications to the varying currents in submarine cables of great interest have been made by Thomson {PhU. Mag., 1856) and Kirchhoff {Pogg. Ann., 1857), while practical researches of the greatest importance to telegraphy have been made on this and kindred subjects by Faraday, Wheatstone, Guillemin, Varley, Jenkin, and others.

Great improvements in galvanometers and galvanometry have been made in our time. One of the first to use an electro-magnetic instrument for measuring or indicating currents was Schweigger, who in 1820 invented the “multiplier.” Nobili used (1825) the astatic “multiplier” with two needles, which is sometimes named after him. Becquerel (1837) used the electromagnetic balance, which was employed in an improved form by Lenz and Jacobi. Pouillet invented the sine and tangent compasses (1837). The defects of the latter instrument were pointed out by Poggendorff, and remedies suggested by him as well as Wheatstone and others. Weber effected great improvements in the construction and use of galvanometers, adapted them for the measurement of transient currents, and elaborated the method of oscillations which h⅜d been much used by Fechner. In 1849 Helmholtz invented the tangent compass with two coils which bears his name. Great improvements in delicacy and promptness of action have been made by Sir William Thomson in galvanometers destined for the measurement of resistance, and for indicating the feeble currents of submarine cables.

The measurement of resistance has been carried to great perfection, chiefly owing to the labours of those who have busied themselves in perfecting the electric telegraph. Among such the highest place must be assigned to Sir Charles Wheatstone; his memoirs in the Philosophical Transactions (1843) gave a great impulse to this department of our science. He invented the rheostat, which underwent several modifications, but is now superseded by the resistance box which was first used by Siemens. The earlier methods of Ohm, Wheatstone, and others for measuring resistance were defective, because they depended on the constancy of the battery which furnished the current. These defects are completely obviated in the more modern “null methods,” which may be divided into two classes—those which depend on the use of the differential galvanometer introduced by Becquerel, and those which are modifications of the Wheatstone’s bridge method, invented by Christie and brought into use by Wheatstone. As examples of the latter, we may mention the methods of Thomson, and of Matthiessen and Hockin, for measuring small resistances, and Thomson’s method for measuring the resistance of the galvanometer (see Maxwell’s Electricity and Magnetism, pp. 404, 410) Many determinations of the specific resistances of metals and alloys have been made by Davy, Ohm, Becquerel, Matthiessen, and others. To Matthiessen in particular science is indebted for great improvements in method and a large body of valuable results in this department. The metals have been arranged in a series according to their conducting powers; and this series is found to be nearly the same for electricity as for heat. The conductivity of metals decreases as the temperature increases, the rate of decrease being nearly the same for most pure metals, but much smaller and more variable for alloys, which, on the other hand, have in general a large specific resistance. The earlier attempts to measure the resistance of electrolytes were not satisfactory, owing to insufficient allowance for polarization. In later times this difficulty has been overcome or avoided, and concordant results have been obtained by Beetz, Paalzow, Kohlrausch, Nippoldt, and Grotrian. The three last, using the electrodynamometer and sine inductor, have made elaborate researches, establishing among many other interesting results that the conductivity of electrolytes increases with the temperature (Pogg· Ann., 1869-74).

The measurement of the electromotive force and that of internal resistance of batteries in action are problems which, in their most general form, are inextricably connected. It is easy to measure with considerable accuracy the electromotive force of an open battery. We have merely to connect its poles with a Thomson’s electrometer, and compare the deflection thus obtained with that due to some standard electromotive force. Another very satisfactory method is Latimer Clarke’s modification of Poggendorff’s compensation method (see Maxwell, 413). It is likewise not difficult to measure by a variety of methods, the most satisfactory being that of Mance (Maxwell, 411), the internal resistance of a battery when it is only traversed by a feeble current. But the measurement of the electromotive force and internal resistance of a battery working a strong current has hardly as yet been achieved with success; not that we undervalue the ingenious and important methods of Paalzow, Von Waltenhofen, Beetz (Wiedemann, i. § 181), and Siemens (Pogg. Ann., 1874). The concordant results of the last two are indeed very remarkable. Still all these methods are more or less affected by the fact that the electromotive force of a battery depends on the current which it is sending (see Beetz in Pogg. Ann., cxlii.).

The “crown of cups” of Volta was the parent of a great many other arrangements for the production of voltaic electricity. These had for their end either compactness or diminution of the internal resistance by enlarging the plates; we may mention the batteries of Cruickshank (1801), Wollaston (1815), and Hare (1822). In 1830 Sturgeon introduced the capital improvement of amalgamating the zinc plates. In 1840 Smee used platinum or silver plates instead of copper; by platinizing these he avoided to a considerable extent polarization by adhering hydrogen. In 1836 Daniell invented the two-fluid battery which bears his name. This battery is the best constant battery hitherto invented, and is, under various modifications, largely used in practical and scientific work. In the same year Grove invented his well-known battery, which surpasses Daniell’s in smallness of internal resistance and in electromotive force, although, on the other hand, it is more troublesome to manage and is unsuited for long-continued action. Cooper, in 1840, replaced the expensive platinum plates of Grove’s battery by carbon. This modification was introduced in a practical form into the battery of Bunsen (1842), which is much used on the Continent, and combines to a certain extent the advantages of Grove and Daniell. Among the more recent of one-fluid batteries may be mentioned the bichromate battery of Bunsen and the Léclanché cell. It is impossible here even to allude to all the forms of battery that have been invented. We may, however, in passing notice the gravitation batteries of Meidinger and Varley, and the large tray cell of Sir William Thomson.

Following up the discoveries of Nicholson, Carlisle, Davy, and others, Faraday took up the investigation of the chemical decompositions effected by the electric current. In 1833 he announced his great law of electro-chemical equivalents, which made an epoch in the history of this part of electricity. He recognized and for the first time thoroughly explained the secondary actions which had hitherto masked the essential features of the phenomenon. Faraday’s discovery gave a new measure of the current, and he invented an instrument called the voltameter, which was much used by those who followed out his discoveries. Space fails us to notice in detail the labours of those who verified and developed Faraday’s discovery. De la Rive, Becquerel, Soret, Buff, Beetz, Hittorf, Matteucci, Daniell, Miller, and many others have worked in this field.

Many theories of electrolysis have been given. That of Grotthuss (1805) has been held under various modifications by many physicists; but none of these theories have done more than give us a convenient mode of representing experimental results. Clausius {Pogg. Ann., ci., 1857) has published a remarkable molecular theory of electrolysis, which is free from some of the objections to the views of Grotthuss and his followers.

The advances made in the experimental study of electrolysis reacted on the theory of the galvanic battery. It was now recognized that the cause of the inconstancy of batteries is the opposing electromotive force due to the existence of the products of decomposition at the plates of the battery. Gautherot, in 1802, observed the polarization current from electrodes which had been used for electrolysis. Ritter confirmed his discovery, and constructed on the new principle his secondary pile. Ohm also experimented on this subject. Fechner and Poggendorff suspected the existence of a transition resistance lfjebergangsιviderstand') at the places where the chemical products were evolved. But the experiments of Lenz, Beetz, and others soon showed that a vera causa existed in the electromotive force of polarization amply sufficient to explaiu their results. The influence of the strength of the current, the size and nature of the plates, time, &c., on polarization have been investigated by many physicists, among whom are promineut Beetz and Poggendorff. Determinations of the electromotive force of polarization have been made by Daniell, Wheatstone, Poggendorff, and Beetz, and recently by Tait and others. Among recent labours on polarization are to be mentioned those of Helmholtz and his pupils. We must not omit to notice here the gas battery of Grove, and the powerful secondary piles which have recently been constructed by Planté. We refer those interested in these and kindred subjects to the exhaustive accounts in Wiedemann’s Galvanismus. Justice to all contributors to our knowledge is impossible in our limited space.

This is perhaps the place to mention the great battle that raged so long between the upholders of the two rival theories of the action of the pile. Volta and his immediate successors held that the current was due to the electromotive force of contact between the dissimilar metals in the circuit, the function of the electrolyte being simply to transmit the electricity, there being no contact force between metals and liquids. The upholders of the chemical theory sought for the origin of the current in the chemical affinity between the zinc and the acid or their equivalents in the battery, and, in the first instance at least, denied the existence of the contact force of Volta. It was soon shown, however, on the one hand, that there was a contact force between metals and liquids, and, on the other, that an electric current could be generated without a heterogeneous metallic circuit at all.

Later holders of both theories modified their views as experiment established the necessity for so doing. Ohm and Fechner and other Continental philosophers inclined to a modified contact theory, and Sir William Thomson at present lends his weighty authority to that side. On the other side are the great names of Faraday, Becquerel, and De la Rive. The contact theorists devoted their attention more to the electrostatic phenomena of the pile, while the chemical theorists studied with great minuteness the phenomena of electrolysis, so that both theories have rendered good service to science. Now-a-days most physicists probably recognize too well the defects of both theories to think it worth while to attack either, and take refuge more or less in eclecticism.

There was one point which the older adherents of the contact theory overlooked, the importance of which was more or less dimly perceived by their chemical opponents. This was, in modern language, the question, where does the energy come from which appears as kinetic energy in the moving parts of electromagnetic engines, as heat in the conducting wires, through which a current is being driven,and so forth? It was not until the dynamical theory of heat had been perfected that the first answer to this question was given. Joule {PhH, Mag., 1841) had arrived experimentally at the law which regulates the generation of heat in conductors by the electric current, and his law was verified by Lenz and Becquerel, both for metals and electrolytes. Reasoning from Joule’s law on the case where the whole of the energy appears in the form of heat, Thomson {Phil. Mag., 1851) established the important theorem that the electromotive force of an electro-chemical apparatus is, in absolute measure, equal to the mechanical equivalent of the chemical action on one electro-chemical equivalent of the substance. Calculations of the electromotive force of a Daniell’s cell, from the results of Joule, Andrews, and Favre and Silbermann, have given numbers agreeing with thedirect measurements of Bosscha. The total amount of the electromotive force in the circuit having been thus satisfactorily determined, the question between the rival theories is reduced to the determination of the seat of this force—At which of the junctions does it act?

Besides his great services in other branches of electricity, Faraday did much to advance electrostatics. His experimental investigations on electrostatic induction are of great interest, and his discovery, of the effect of the medium between the electrified bodies opened out a new aspect of the phenomenon quite unsuspected by those who held too closely to the theories of action at a distance. He introduced the term specific inductive capacity, and measured the capacity of several solid substances, showing that in these it was much greater than that of air. He conceived that his results were at variance with any theory of action at a distance, and gave a theory of his own, which accounted for all his facts, and which guided him in his investigations. Matteucci and Siemens adopted the views of Faraday, and the latter introduced refined methods for measuring specific inductive capacities. Such measurements have been made in later times by Barclay and Gibson for paraffin, and by Silow for certain fluids. The most remarkable result thus obtained, however, are those of Boltzmann, who succeeded not only in detecting but in actually measuring the differences between the specific inductive capacities of different gases. Faraday had looked in vain for such differences, and concluded that the specific inductive capacity was the same for all gases. The phenomenon of the residual discharge was recognized and experimented on by Faraday. Kohlrausch, Gaugain, Wüllner, and others have also experimented on it; and quite recently Mr Hopkinson has obtained some very interesting results regarding the superposition of residual discharges. These results are analogous to the curious phenomena of “elastic recovery” observed by Kohlrausch.

Sir W. Snow Harris was a very able experimenter, and did much to improve electrostatical apparatus. He used the electrical balance and the bifilar suspension balance invented by himself. On the strength of his results he questioned the soundness of the views of Coulomb. The work of Harris on the influence of the surrounding medium on the electric spark is of great importance. Faraday made a series of beautiful experiments on this subject, and arrived at a body of results which still form a good portion of the established facts on this subject. Very important in this connection are the measurements of Sir W. Thomson of the electromotive force required to produce a spark in aiι∙ between two conductors, which he has found to be disproportionally smaller for large distances than for small.

The luminous phenomena attending the elecfric discharge, especially in vacuum tubes such as those of Geissler, are exceedingly beautiful, and have of late formed a favourite subject of experimental study. Many interesting results have been obtained, the significance of which we may not yet rightly comprehend. Among the older labours in this field we may mention those of Pliicker and Hittorf, De la Rive, Riess, Gassiot, and Varley. But even as we write cur knowledge of the subject is extending, and we refrain from referring to more modern results; for historical sketching—a difficult task in any case—is unsafe in an open field like this, where some apparently insignificant fact may contain the germ of a great discovery. We may here mention the experiments of Wheatstone on the velocity of electricity, valuable less for the results he obtained than for the ingenious application of the rotating mirror, then used for the first time, which has since been applied with much success in the study of the electric discharge.

One of the greatest names in electrical science is that of Riess. In his classical research on the heating of wires by the discharge from a battery of Leyden jars, he did for electricity of high potential what Joule did for the voltaic current. The electro-thermometer which he used in these researches was an improvement on the older instruments of Kinnersley and Harris. Riess repeated and extended the experiments of Coulomb, and effected many improvements in the apparatus for electrostatical experiments. His Reibungs-electricitat is a work of great value, and was for long the best book of reference open to the experimental student. Happily we have now another in the recently published work of Μ. Mascart.

Sir William Thomson revolutionized experimental electricity by introducing instruments of precision. Chief among these are his quadrant and absolute electrometers. His portable electrometer and water-dropping apparatus are instruments of great value to the meteorologist in the study of atmospheric electricity, a science which he has done much in other ways to forward. Besides this, we owe to him many valuable suggestions for electrical apparatus and experimental methods, some of which have been carried out by his pupils.

The theory of statical electricity has made great progress since Poisson’s time. Among its successful cultivators we may mention Murphy {Electricity, 1833), and Plana (1845). The latter went over much the same ground as Poisson, extending his results. It was, however, by Green {Essay on The Application of Mathematical Analysis to the Theories of Electricity and Magnetism, 1828; or Mathematical Papers, edited by N. M. Ferrers), a self-taught mathematician, that the greatest advances were made in the mathematical theory of electricity. “His researches,” as Sir William Thomson has observed, “have led to the elementary proposition which must constitute the legitimate foundation of every perfect mathematical structure that is to be made from the materials furnished in the experimental laws of Coulomb. Not only do they afford a natural and complete explanation of the beautiful quantitative experiments which have been so interesting at all times to practical electricians, but they suggest to the mathematician the simplest and most powerful methods of dealing with problems which, if attacked by the mere force of the old analysis, must have remained for ever unsolved.” One of the simplest applications of these theorems was to perfect the theory of the Leyden phial, a result which (if we except the peculiar action of the insulating solid medium, since discovered by laraday) we owe to his genius. He has also shown how an infinite number of forms of conductors may be invented, so that the distribution of electricity in equilibrium on each may be expressible in finite algebraical terms,—an immense stride in the science, when we consider that the distribution of electricity on a single spherical conductor, an uninfluenced ellipsoidal conductor, and two spheres mutually influencing one another, were the only cases solved by Poisson, and indeed the only cases conceived to be solvable by mathematical writers. The work of Green, which con tained these fine researches, though published in 1828, had escaped the notice not only of foreign, but even of British mathematicians; and it is a singular fact in the history of science that all his general theorems were rediscovered by Sir William Thomson, Chasles and Sturm, and Gauss (see Reprint of Thomson’s papers). Sir William Thomson, however, pushed his researches much further than his fellow-labourers. He showed that the experimental results of Sir William Snow Harris, which their author had supposed to be adverse to the theory of Coulomb, were really in strict accordance with that theory in all cases where they were sufficiently simple to be submitted to calculation. He was guided in his earlier investigations by an analogy between the problems involved in steady flux of heat and the equilibrium of electricity on conductors. He showed in 1845 how the peculiar electric polarization discovered by Faraday in dielectrics, or solid insulators subjected to electric force, is to be taken into account in the theory of the Leyden iar, so as to supply the deficiency in Green’s investigations. We also owe to Sir William Thomson new synthetical methods of great elegance and power. The theory of electric images, and the method of electric inversion founded thereon, constitute the greatest advance in the mathematical theory of electrostatics since the famous memoir of Green. These he has applied in the happiest manner to the demonstration of propositions which had hitherto required the resources of the higher analysis, and he has also found by means of them the distribution on a spherical bowl, a case of great interest in the theory of partially closed conductors, which had never been attacked or even dreamt of as solvable before. The work of Professor Clerk Maxwell on Electricity and Magnetism, which appeared in 1873, has already exerted great influence on the study of electricity both in England and on the Continent. In it are fully given his valuable theory of the action of the dielectric medium. He regards the electrical forces as the result of stress in the medium, and calculates the stress components which will give the observed forces, and at the same time account for the equilibrium of the medium. The striking discovery recently made by Mr Kerr of Glasgow, of the effect on polarized light exerted by a piece of glass under the action of strong electric force, is of great importance in connection with Maxwell’s theory, and realizes a cherished expectation of Faraday, of whom Maxwell is the professed exponent. We must allude here once more to Maxwell’s electromagnetic theory of light, the touchstone of which is the proposition that in transparent media, whose magnetic inductive capacity is very nearly equal to that of air, the dielectric capacity is equal to the square of the index of refraction for light of infinite wave length. Although, as perhaps was to be expected, owing to disturbing influences such as heterogeneity, this proposition has not been found in good agreement with experiment in the case of solids, yet for liquids (Silow, Pogg. Ann., civ. clviii.) and gases (Boltzmann, Ibid. civ.) the agreement is so good as to lead us to think that the theory contains a great part of the whole truth.

In the earlier stages of the science several units were in-, troduced for the measurement of quantities dealt with in electricity. As examples of these we may mention the wire of Jacobi, and the mercury column of Siemens, a metre long, with a section of a square millimetre, which at given temperatures furnished units of resistance; the Daniell’s cell, which furnished the unit of electromotive force, the chemical unit of current intensity, &c. All these units were perfectly arbitrary, and there was no connection of any kind between them. The introduction of a rational system of unitation, based on the fundamental units of time, mass, and length, was one of the greatest steps of our time. The impulse came from the famous memoir of Gauss, Intensitas Vis Magneticae Terrestris ad Mensuram absolutam revocata, 1832. In conjunction with Weber, he introduced his principles into the measurement of the earth’s magnetic force. To Weber belongs the credit of doing a similar service for electricity. He not only devised three different systems of such units—the electrodynamical, the electrostatical, and the electromagnetic—but he carried out a series of measurements which practically introduced the last two systems. The fundamental research in this subject is to determine in electromagnetic measure the resistance of some wire from which, by comparison, the electromagnetic unit of resistance can be constructed. Measurements of this kind were made by Kirchhoff in 1849; more carefully in two different ways by Weber in 1851; by the committee of the British Association in 1863, Arc.; by Kohlrausch in 1870; and by Lorenz in 1873. Accounts of these important researches will be found in Wiedemann and Maxwell, and in the collected reports of the British Association on “Electrical Standards.” The ratio of the electrostatic to the electromagnetic unit of electric quantity is a velocity (according to Maxwell’s electromagnetic theory of light it is the velocity of light), the experimental determination of which is of the greatest theoretical and practical importance. Such determinations have been made by Weber and Kohlrausch in 1856, by Maxwell in 1868, and by Thomson in 1869. The results are not so concordant as might be desired, but the research is a very difficult one.

For convenience in practice the British Association committee have recommended certain multiples of the absolute unit, to which they have given names—e.g., the .Ohm, the Volt, the Farad, &c. These have become current to a great extent among practical electricians in this country. For practical purposes, an empirical standard of electromotive force has been introduced by Latimer Clark, whose value in volts is given as 1 ’ 457. It is very important, in order to be able to reduce chemical to absolute measure, to know accurately the electro-chemical equivalent of water. Values for this have been found by Weber ( 184(5), Bunsen (1843), Casselman (1843), and Joule (1851). Kohlrausch (1873) made a careful determination of the electro-chemical equivalent of silver, from which the electro-chemical equivalent of water can be calculated.

GENERAL SKETCH 0F PHENOMENA.

If a piece of glass and a piece of sealing-wax be each rubbed with a dry woollen cloth, it will be found that both the glass and the wax have acquired the property of attracting indiscriminately any small light body in the neighbourhood; and it will be further observed, in many cases, that the small bodies, after adhering for a little to the glass or wax, will be again repelled.

These actions have at first sight a likeness to the attractions and repulsions of magnetic bodies, but they are sufficiently distinguished from these—1st, By their origin,— being excited by friction and other causes in a great variety of bodies, whereas magnetic action is powerfully exhibited and communicated only by certain varieties of iron and iron ore, by nickel and cobalt, and by certain arrangements which we shall have to mention by-and-by; 2d, By the nature of the bodies acted on; for these may be, in the case of excited glass or wax, light particles of any substance, whereas the only bodies powerfully acted on magnetically are either magnets or their equivalents, or iron, nickel, and cobalt; and 3d, By the fact that every magnet has two poles possessing opposite properties, whereas an electrified body may have similar properties in every part of its surface.

If the experiment were carefully tried it would be found that a piece of glass excited as above repels another piece of glass similarly excited, but attracts an excited piece of wax. A convenient way of exhibiting these actions, which also brings under our notice another fact of fundamental importance, is as follows. Two gilt balls of elder pith are fastened to the ends of a light needle of shellac, which is balanced horizontally on a point carried on a vertical stand (fig. 1). To the stand a stop is fixed for convenience, to prevent the needle from spinning more than half round. If we touch the ball A with a piece of excited glass, and B with a piece of excited sealing-wax, and touch a ball C, fastened to a shellac stem, with a piece of excited glass, then C will chase A away till it is brought up by the stop, while it will, on the other hand, attract B. If, again, C be touched with a piece of excited wax, it will attract A and repel B.

Pieces of glass or wax excited in this way are said to be electrified, and the balls which by contact have acquired properties similar to those of the originally electrified bodies are said to be electrified by conduction.

It appears from the above experiment that the electrifications of glass and sealing-wax, when rubbed with woollen, have opposite properties, which they communicate to bodies brought into contact with them. A body which has similar electrification to a piece of glass rubbed with woollen is said to be vitreously or positively electrified; a body with similar electrification to a piece of sealing-wax rubbed with woollen is said to be resinously or negatively electrified. The result of the above experiment may then be summarized thus :—

Bodies similarly electrified, whether positively or negatively, repel each other.

Bodies oppositely electrified attract each other.

We have seen that a pith ball becomes, by contact with a positively electrified piece of glass, itself positively electrified. If we take two pith balls, electrify one of them positively, and then touch both simultaneously by a piece of thin wire, suspended by white silk, and test them with the electroscopic needle described above, they will be found both positively electrified; each will repel A and attract B, though less powerfully than the originally electrified ball did, before the connection between them was made. The success of the experiment will be found independent of the length or shape of the wire, and will be equally good with silver, gold, iron, lead, or any other metal. But, if we use a thread of glass or shellac to connect the balls, the electrification of the first ball will be found unaltered, and the second will remain neutral—that is, it will not attract or repel another neutral ball, and will equally attract both balls, A and B, of the electroscopic needle. The difference in the power of transmitting electrical properties from one body to another, or of aiding in electrification by conduction, leads us to divide all substances into two classes—conductors, which do very readily, and non-conductors, which do not, or do not very readily, transmit electrification from one body to another. If we connect an electrified conductor by means of another conductor to a very large conducting body, such as the earth, it will be found that so much electrification has been carried away from the small body that it is left sen sibly neutral. If, accordingly, we wish a conducting body to preserve its electrification unaltered, we must support it on some non-conducting substance. When thus supported the body is said to be insulated, the non-conducting support being called the insulator, a name which has on that account been given to non-conductors generally.

We have remarked above that a neutral pith ball attracts equally the positive and negative balls of the electroscoρic needle; this leads us to remark, more explicitly than we have hitherto done, that an electrified body in general and in the first instance attracts a neutral or unelectrified body. The explanation of this action is that the originally neutral body in presence of the electri tied body becomes itself electrified for the time. It is said to be electrified by induction, and it is very easy to show, by using large bodies, not only that the originally neutral body is actually electrified, but that it is oppositely electrified in different parts. Thus (fig. 2) A and B are two bodies suitably insulated and placed one above the other. If B be originally neutral, and A be positively electrified, then the lower end of B will be negatively, and the upper end positively electrified; as may be easily shown by exploring with a small positively electrified pith ball suspended by a dry white silk thread; the little ball will be attracted towards the lower end of B, and repelled from the upper. If we remove the body A, or, which (as we have seen) amounts to the same thing, connect it with the earth, and so “discharge” its electrification, we shall find that all traces of electrical action in B have disappeared—i.e., the small positively electrified pith ball will be attracted everywhere; and, if we discharge it too, it will neither be attracted nor repelled anywhere.

Provisional Theory.

Before going further into detail, it will be convenient to give a working theory of electrical phenomena, so far as we have considered them. The use of such a theory at the present stage is to enable us to co-ordinate and classify the results of experiment, and to furnish a few leading principles under which we may group results which appear to be due to a common cause. Such ' a theory is invaluable as a memoria technica for experimental results, and is useful in suggesting directions for experimental inquiry; but in framing it we must be careful to make it contain as little as possible beyond the results of actual experiment, and in using it we must be on our guard aga⅛st allowing it to prepossess our minds as to what may be the ultimate explanation of the phenomena we are considering.

Following the caution of Coulomb and the example of Sir William Thomson, we shall avoid the use of the term electrical fluid, and substitute instead the more succinct and less misleading word electricity. We suppose that a body which exhibits electrical properties (as above defined) has associated with its mass a certain quantity of something which, without attempting further definition, we shall call electricity. Of our right to use the word quantity here we shall give experimental justification by-and-by, and then the question of the appropriate unit will (vide infra, “electric quantity”) be discussed. We may suppose that elec tricity is distributed throughout the whole mass of a body, and speak of electrical “volume density,” meaning the quantity of electricity in an element of volume divided by the element of volume. We shall also speak of an element of electricity, meaning the electricity in an element or very small portion of a body. In certain cases we shall find that electricity resides on the surface of a body; electrical “surface density” then means quantity of electricity on an element of surface divided by the element of surface, and element of electricity the electricity on an element of surface.

For shortness, we shall denote positive or vitreous electricity by the mathematical sign + , and resinous or negative electricity by the sign —, remarking that the choice of the signs is arbitrary, and reserving for the present the question of how far we may associate with thèse signs the corresponding mathematical ideas.

We shall assume that every element of electricity repels every other element of the same sign, and attracts every other element of opposite sign. The precise law of this force will be investigated further on

This force considered as acting on any element of electricity we shall call an electric force. In perfectly conducting substances electricity moves with perfect freedom under any electromotive force, however small. In perfect non-conducting substances electricity will not move under any electromotive force, however great. Any case in nature lies somewhere between these extremes, but into questions of gradation, &c., we do not enter for the present.

When the forces due to other electrical elements acting on the electricity jn any element of a body have a resultant, that resultant acts on the element itself, and is called the ponderomotive force, to distinguish it from the electromotive (or electric) force which tends to move + electricity in one direction and - electricity in the opposite direction.

When a body is neutral, we shall assume that it contains equal and equally distributed quantities of + and — electricity, and we shall further suppose those to be practically unlimited in amount. A + electrified body is then to be conceived as a body w hich has excess of + electricity and a - electrified body as one which has excess of - electricity. Communication of + electricity to a body is in accordance with this to be regarded as equivalent to the abstraction of an equal amount of - electricity, and conversely.

It is easy to see that the above assumptions will explain in a general way the phenomena already described. Thus the + electricity of the electrified pith ball C acting bn the + electricity of the ball A of the electroscopic needle repels it, and this force by our assumption is equally exerted on the matter of A, therefore A tends to move away from C, and will do so as long as it is free to move. The action on the - electrified ball B is similarly explained. Conduction and discharge to earth may be explained in a similar manner.

The attraction of an electrified body ( ÷ let us suppose) A on a neutral insulated body B is thus explained. The + electricity on A (fig. 3) attracts the - electricity in B and repels the + electricity, so that, though there is still on the whole as much + electricity as - electricity, yet the distribution is no longer the same, for, the electricity being free to move, the - electricity under the attraction approaches A until the non-conducting air and the attraction of the separated + electricity on B stops it, and the + electricity recedes in similar fashion. When electrical equilibrium has been attained the action of the + electricity of A on the - electricity of B will exceed its action on the + electricity of B, which is on the whole more distant, the electromotive force on the electricity of B will be on the whole attractive, and hence the pondero-ιnotive force on B, will be also attractive.

The above explanation involves of course the general explanation of electrification by induction.

Experimental investigation of Electrical Quantity, Distribution, and Force.

In what follows we shall suppose that we have an instrument which will serve as an electroscope and to some extent as an electrometer; that is, which shall tell us readily whether a body brought into communication with it is + or - electrified or not at all, and also enable us to tell when one body is more strongly electrified + or - than another.

The gold-leaf electroscope of Bennet or the dry pile electroscope of Bohnenberger will meet these requirements, and have been much used in electrical researches. We shall, however, suppose that we are using the rudimentary form of Thomson’s electrometer constructed by Elliot Brothers for lecture-room experiments, which is now much used in England, and answers very well. For a description of these and other electroscopes and electrometers, see article Electrometer.

We shall also assume for the present that we have the means of producing and communicating to any body as much of either kind of electrification as we please, and pass on to consider the data of experiment regarding the distribution of statical electricity in conducting bodies. We are thus at the very outset brought face to face with the idea of electric quantity.

Electric Quantity.

We have to explain how the introduction of the term quantity into electrical science is justified by experiment, and how we can multiply and subdivide quantities of electricity. Although it Î3 no doubt possible to introduce the notion of quantity independently of the measure of electric force, yet the most convenient and practical measure of quantity depends on the measurement of force, and the absolute electrostatic unit of quantity is stated in this way. We are naturally led, therefore, to combine with the study of quant'ty and distribution the experimental study of the laws of electric force.

We shall have occasion to allude to two leading experimental methods that have been used in investigating the present subject. These might be called the old method and the new.

The old method, which did so much for electrical science in the master hand of Coulomb, depended on the use of the torsion balance and proof plane, both invented by Coulomb himself. This method was used by Reiss and others up to Faraday’s time.

Michell, about Coulomb’s time or a little before, first suggested the idea of measuring small forces by the torsion of a wire. He proposed to apply the method to measure the attraction of gravitation between two bodies of moderate size, thus finding the mean density of the earth, and the method was actually carried out by Cavendish; but Coulomb was in all probability unaware of Michell’s suggestion. He made careful preliminary experiments (the first of the kind) on the torsion of wires, and found that the couple required to twist a straight wire through a given angle varies as the angle of torsion multiplied by the fourth power of the diameter of the wire directly, and as the length of the wire inversely (Mem. de l ’ Acad., 1784).

The balance used by Coulomb in most of his experiments is represented in figure 4.

ABDC is a cylinder of glass 1 foot in diameter and 1 foot high. This cylinder is closed by a glass lid pierced centrically and eccentrically by two openings, each about 20 lines wide. Into the middle opening is cemented a glass tube 2 feet high, to the upper end of which is fitted a torsion head; the separate parts of the head are shown larger at the side of the figure. H is a collar cemented to the glass tube; MO a metal disc, divided on the edge into 360 degrees; this disc is fastened to a tube N, which slips into the collar H. K is a button whose neck turns easily in a hole in MO; to the lower part of the button is fastened a small clamp, which seizes the wire of the balance. I is an arm with a small projecting piece which slips over the edge of the disc MO.

This piece has a fiducial mark on it, which enables us to read off the position of the arm on the graduated edge of MO. The horizontal arm bd consists of a silk thread or fine straw covered with sealing wax terminated by a thread of shellac at b about 18 lines long, which carries a pith ball 2 or 3 lines in diameter. At the other end of the arm is a vertical disc of oiled paper, which serves as a counterpoise to the pith ball, as a damper to the oscillations, and as an index by means of which the position of the horizontal arm can be read off on a graduation carried round the glass cylinder. The eccentric hole in the cover of the balance allows the introduction of the fixed ball α; this is carried on a shellac stem fastened to a clamp P, which by means of fiducial marks can be placed in a fixed position on the cover. The wire in Coulomb’s balance was of silver, about 30 cm. long. Its diameter was Ό035 cm., and it weighed about - 003 gm. He found by the method of oscillations that a couple equivalent to the weight of T7 milligramme, acting at the end of an arm a decimetre long, would keep the wire twisted through 360°.

Besides this form of balance Coulomb used others, some more delicate for electroscopic purposes, and others less so, but of larger dimensions, into which he could introduce electrified bodies of considerable size.

Faraday used Coulomb’s balance, and Snow Harris used the bifilar balance, which is a modification of Coulomb’s. In the second volume of his Experimental Pesearches, however, Faraday gives a general method of experimenting, which to a great extent has superseded the older method. This may be called the “cage method;” it depends for its success on the use of some delicate instrument for measuring differences of potential; this was supplied by the quadrant electrometer of Sir William Thomson, which has thus completely revolutionized the whole system of electrostatic measurement. . .

Faraday’s experiment was as follows (Exp. Pes., vol. ii. p. 279)

Let A (fig. 5) be an insulated hollow conductor with an opening C to allow admission to the interior. Faraday used a pewter ice pail, α 10⅛ in. high and 7 in. in diameter Connect the outside of A with one electrode of an electrometer E, which may for most purposes be the rudimentary form of Thomson’s electrometer mentioned above. Connect the other electrode of the electrometer with the earth. If now we introduce a positively electrified body, say a "brass ball C, suspended by a white silk string, we shall find that the electrometer needle is deflected through a certain angle, the spot of light going a certain distance to the right, say, of the scale. It will be found that, provided the ball C is more than a certain depth (about 3 in. in Farady’s experiment) below the mouth of the pail, no further motion of the ball, right or left, up or down, will affect the indications of the electrometer. It will also be found that the same indications will be got to whatever point of the outside of the paiJ the electrometer wire is attached. If we dimmish or increase the + electrification of C, the electrometer deflection will diminish or increase accordingly. If we introduce a negatively electrified ball C', the deflection will be to the left, and everything else as before. If C gives a certain positive (right) deflection, and C' an equal (left) deflection, then if we introduce C and C' together, the deflection will be zero. If C and C' be both + electrified and give equal + deflections, then introduced together they will give a double + deflection, and if three such balls, all giving equal + deflections, be introduced together, they will give a treble + deflection.

It is obvious that this experiment of Faraday’s not only gives a very ready test of the electrical state of bodies, but at once suggests the notion of electrical quantity, and a theoretically possible electrostatic unit. Suppose, in fact, we take for our test the deflectiou of a Thomson’s electrometer of given sensibility, then we might specify as a unit of electrical quantity the quantity of + electricity on or in a brass ball of given size, which will produce with a given cage a certain given deflection of the electrometer.

To make this definition useful we must have the means of transferring a given charge from one body to another, and charging a body with any multiple or submultiple of our unit, and of charging a body with any multiple or submultiple of the unit of negative electricity, which we may define as the quantity of — electricity which will just annul the action of the unit of + electricity in the electric cage.

All these requirements may be satisfied by suitably modifying Faraday’s experiment.

We saw that we might move the ball about in the middle of our electric cage without affecting the electrometer deflection; we find, moreover, that when we withdraw the electrified ball without touching the cage, the needle returns to zero. If, however, before withdrawing the ball we cause it to touch the inside of the cage, the electrometer deflection remains the same as before, and after the ball has been removed the deflection is still the same, while if we examine the ball, we find that all traces of electrification have disappeared.

To transfer a given quantity of electricity.—If we pro vide ourselves with two cages, a large one G, and a small one H, and take a ball C, electrified positively with unit quantity as above defined, then testing C in cage G, in connection with the electrometer, we get a certain deflection D. If now we transfer the electrification of C to H, by the process just described, and then put H inside G, we shall get the same deflection I) as before. It appears, therefore, that we can transfer electrification from one body to another without loss; we thus fulfil one of our requirements, and give an additional justification of the use of the word quantity in the present case.

To get any multiple or submultiple of the electric unit.— We may repeat the process above performed on the small cage H by touching its inside with the ball C, again electrified to unit quantity. All the electrification will pass to H

as before, and if we now test 11 in G we shall get a deflection 2D. We can in this way get any multiple we please of the unit charge. If we take the electrified brass ball C and touch it by a perfectly equal neutral ball C', on introducing C into G we shall get deflection ⅜ D; if we touch C again by C', previously rendered neutral, we shall get deflection ∣ D, and so on; if we had touched C simultaneously, as in fig. 6, with two equal neutral balls, we should have got deflection ⅜ D, aud so on. We can thus get any submultiple of our unit charge.

To get a given multiple and submultiple of the negative unit.—This is possible when we can get a quantity of — electricity, which will just destroy the action of a given quantity of + electricity in the electric cage. If we introduce our given quantity of + electricity into the cage H, without allowing the conductor carrying it to touch the cage and at the same time put the outside of the cage in communication with the ground, then if we remove the conductor with the given quantity of + electricity and put it in G, it will give the same + deflection as before, while H tested in the same way will give a negative deflection exactly equal to the former, and if both be introduced together there will be no deflection. We can, therefore, in this way get a - quantity equal aud opposite to a given + quantity.

Electrical Distribution.

Experiments had been made before Coulomb’s time to determine what effect the nature of a body has on electrical distribution. Gray and White concluded, from an experiment with two cubes of oak, one hollow and the other solid, “that it was the surface of the cubes only which attracted.” Le Monnier showed that a sheet of lead gave a better spark when extended than when rolled together. These experiments point to the conclusion that electrical distribution in conducting bodies depends merely on the shape of the bounding surface.

We may make experiments confirmatory of this conclusion with the electric cage. If we electrify a brass sphere A, and then touch it with another sphere B, and test the electrification of B in the cage, we shall find that the amount of electricity taken by B is always the same, whatever its material may be, so long as the radius of its external surface is the same. Experiment is unable to detect any difference in this respect between a solid sphere of lead and the thinnest soap-bubble of the same radius. Coulomb took a large cylinder of wood, in which he made several holes four lines in diameter and four lines deep. Having electrified the cylinder and insulated it, he examined its electrical condition by means of the proof-plane. This instrument, so much used by Coulomb, consisted merely of a small disc of gilt paper (in this case a line and a half in diameter) fastened to the end of a needle of shellac. The disc is applied to any point of a body whose electrification we wish to test so as to be in the tangent plane to the surface of the body. Assuming for a moment, what we shall by-and-by prove, that electricity resides on the surface of bodies, it is natural to suppose that the proofplane, when placed as described, will form part of the bounding surface, and will therefore take up as much electricity as was originally on the part of the surface which it covers. If now we remove the proof-plane in the direction of the normal, keeping it, as nearly as possible, parallel to the surface, the electricity will not leave it; but we shall carry safely away all that it had when in contact with the surface of the body. We may return to the consideration of the proof-plane when we come to study mathematically the laws of electrical distribution.

In the experiment with which we are now concerned, Coulomb used a very delicate balance (a force ∩f 77⅛y of a milligramme was sufficient to keep the wire twisted through 90°). When the proof-plane was applied to any point of the external surface of the wooden cylinder, and then introduced into the torsion balance, it repelled the electrified ball of the balance with great force. When it was carefully introduced into one of the holes, made to touch the bottom, and then carefully withdrawn so as not to touch the edge of the hole, it produced no appreciable effect on the balance.

Coulomb varied this experiment as follows. He insulated and electrified a hollow sphere of metal (fig. 7), and by carefully introducing a proof-plane through a small opening tested the electrical condition of the interior surface. He found no sensible trace of electricity inside, except very near the edge of the small opening. Hence we conclude that if the sphere had been closed entirely there would have been no electrification inside. Many experiments have been made to illustrate the proposition that electricity resides entirely on the surface of conductors. Franklin put a long chain inside a metal teapot, which he insulated aud electrified. When he seized the chain by a hook at the end of a glass rod and pulled it out of the teapot he observed that a pair of pith balls, suspended side by side from the teapot, collapsed more and more as the chain was drawn out, and he concluded that the electrification of the teapot, being now spread over a greater surface, had become weaker.

The following experiment of Biot’s has become classical. A spherical conductor A (fig. 8) is supported on an insulating stem D. B and C are two hollow hemispheres fastened to insulating handles E and F. When these are fitted together they form a sphere somewhat larger than A, with a small hole in it through which the stem D can pass. If we electrify A very strongly, so that when put in the electric cage it powerfully deflects the electrometer, and then close B and C over A, and make either B or C touch it, then separate B and C, and test A, B, and C in the cage, we shall find that all the electricity has gone from A and spread itself over B and C.

The following is an ingenious experiment of Faraday’s, involving the same principle. AB (fig. 9) is a wire ring supported on an insulating stand; C is a conical muslin bag fitted to the ring with two strings fastened to the vertex of the cone, giving the experimenter the means of quickly turning the bag inside out. If the bag be electrified in the first position in the figure and tested with the proofplane and electric cage, it will be found that the outside only is electrified. If we turn the bag inside out and test it, we shall find as before that what is now the outside, and was formerly the inside, is alone electrified. The electricity has thus passed through the bag so as to be on the outside in both cases.

Before leaving for a time the question of the distribution of electricity on conductors, it may be well to warn the student to accept with due reserve the proposition that electricity resides entirely on the surface of conductors, and to remind him that such a proposition is from the nature of the case incapable of direct experimental proof, for we cannot operate in the substance of a mass of metal. Some of the experiments we have quoted bear more directly on the question than others. Coulomb’s experiment, for instance, shows, strictly speaking, merely that electricity avoids cavities and re-entrant angles. Again, in Faraday’s experiment with the linen bag, we have not clearly defined what we mean by the outside of the body. The proposition has on the whole been suggested rather than proved. Its meaning will become clearer as we go more and more into the theory of distribution, and we shall meet with it by-and-by as one of the first propositions in the mathematical theory.

Laιvs of Electric Force.

Before proceeding to give an account of Coulomb’s quantitative experiments on electrical distribution, we shall describe shortly the methods by which he arrived at the laws of electric force, and did for electricity what Newton did for astronomy, i.e., laid the foundation for a mathematical theory of the subject based on the hypothesis of action at a distance.

In this research Coulomb used the form of balance which we described above. The clamp holding the fixed ball of the balance is so adjusted that the centre of the ball falls in a horizontal line through zero of the graduation on the glass cylinder and the prolongation of the suspending wire; the torsion button is turned till its arm is at zero; the disc, button and all, is then turned till the disc on the arm and the centre of the movable ball are in a line with the zero of the lower graduation. The fixed ball, which had been removed to allow of the last adjustment, being replaced, and the movable ball having come to rest in contact with it, both are electrified by means of a small metal ball carried on an insulating stem of shellac. The balls repel each other, and the movable ball takes up a certain position of equilibrium; the corresponding angle is read off. The torsion button is then turned through an angle which Î3 noted, so as to bring the balls nearer together. The new position of the beam is again read off; this may be repeated a third time. We are then in possession of data from which we can draw conclusions as to the law of electrical force at different distances.

Let us assume that the force between two elements of positive electricity (supposed collected at two points, technically speaking, “regarded as physical points”) varies inversely as the square of the distance between them. It will be shown in the mathematical theory that two spheres uniformly electrified, as we shall at present assume the two balle in the balance to be, repel each other, as if their electricity were collected at their centres.

Let e be the angle of equilibrium in any case, τ the angle of torsion. 0 (ßg. 10) is the centre of the beam, F and M the centres of the fixed and movable ball (we suppose OF = OM)j OK is perpendicular to FΜ. Then FM 2 oc sin*

Hence moment of the force on M about os *

0 oc , and the torsonial couple « r + e.

sin 2 ^

Hence in the three cases the value of (r + e) sin ~ tan L = A (say)

2 must be the same, if the law of the inverse square agree with the experiments.

Coulomb made many experiments of the kind we have described. The following is the result which he has given of one such :— [table]

The third column is obtained from the two preceding. A is calculated by putting τ = 0 and e — 36° in the formula

(τ + c) sin ~ tan ~ ~ A .

Then using this value of A and the observed value of τ, the formula is employed to find e in the two second cases. The agreement between the observed and calculated values of e is the test of the truth of the law we have assumed. The agreement in the second line is as good as can be expected when possible errors of experiment are considered. It will be seen, moreover, that the calculated is in excess of the observed value, which is what we ought to expect, owing to the loss of electricity which goes on during the time consumed in the experiment. That there is such a loss may be proved experimentally by simply leaving the movable ball to itself after any of the three operations; it will be seen to move slowly towards the fixed ball. We shall return hereafter to this loss of electricity, with regard to the exact nature of which authorities are not quite agreed. In the third line the agreement is less good, but here the proximity of the balls renders the supposition of uniformity no longer even approximately allowable. The mutual repulsion tends to drive the electricity on each ball farther from the other ball, and thus the action between the balls is as if the electricity on each were collected at points beyond the centre, so that the observed repulsion must be less than that calculated on the supposition of uniformity of distribution.

Coulomb also made experiments with the torsion balance to test whether the law of the inverse square applies to the attraction as well as to the repulsion of electrified bodies. His experiments confirmed the law; but the difficulty of operating is much greater in this case than in the former. He therefore adopted another method of experimenting. A small conducting disc was fixed nor mally on the end of a small shellac needle, which was hung up, so as to be horizontal, on a fibre of raw silk attached to a horizontal scale. Au insulated conducting globe was set up with its centre in the same vertical plane as the scale, and in the same horizontal plane as the centre of the small disc. The globe and disc were oppositely electrified, and the period of oscillation of the needle was found by observing the duration of 15 swings. The time of oscillation follows the pendulum law, and varies inversely as the square root of the force acting on the needle, hence the duration of 15 oscillations will vary inversely as the square root of the force, i.e. directly as the distance between the centres of the globe and disc, if the law of the inverse square hold. Coulomb’s experiment gave the following results :— [table]

The numbers in the third column ought to be all equal. The deviation from equality are not greater than can fairly be explained by loss of electricity and errors of observation.

Coulomb also investigated, both by means of the torsion balance and by the method of oscillations, the relation between electric force and quantity.

He electrified the two balls of the torsion balance by simultaneous contact with another ball, and observed the angle of equilibrium: he then halved the quantity on the fixed ball by touching it with an equal neutral ball, and reduced the torsion till the angle of equilibrium, and, in consequence, the distance between the balls was the same as before; he found the torsional couple in the second case to be somewhat less than half what it was in the first. He therefore concluded that the force between two elements of electricity varies as the product of the quantities.

Coulomb’s experiments were repeated, and his results confirmed by Riess, and by Marié-Davy. Experiments which, when properly interpreted, lead to the same results, were made by Snow Harris, and by Egen.

We have then arrived at this general law of electric force :—

If two quantities q, f of electricity be supposed collected at two points, whose distance is d, the force between them acts in the straight line joining the points and ce ·

d z

So far, this law might be merely an approximation to the truth. Later on, however, it will be seen to be logically deducible from experiments which in delicacy infinitely surpass those just described. The law of Coulomb is in fact established as certainly as the law of gravitation itself.

By means of the law now given the unit of electrical quantity can be defined in a satisfactory and practical manner. This unit we now state to be that quantity of positive electricity which, when collected into a point, repels with unit of force an equal quantity similarly collected into a point at unit distance from the former.

If we take centimetre, gramme, and second as our units of length, mass, and time, the unit force will be that force which in a second generates in a gramme of matter a velocity of a centimetre per second.

The law of electric force between two quantities g and <£ now becomes

Force = ·

The unit of quantity which we have just defined is called the electrostatic unit, in contradistinction to the electromagnetic unit which we shall define hereafter.

Since the dimension of unit of force is [LMT~ 2 ], where L,M,T symbolize units of length, mass, and time, we have for the dimension of unit of electrical quantity [Q]

[q] - [LF i ] = [L i M⅛- ’].

Quantitative Results concerning Distribution.

It has already been indicated that electricity in equilibrium resides on the surface of conducting bodies. We must now review shortly the experimental method by which this surface distribution has been more closely investigated. We shall state here some of the general principles arrived at, and one or two of the results, reserving others for quotation when we come to the mathematical theory of electrical distribution.

The most important experiments are due to Coulomb. He used the proof-plane and ths torsion balance. Riess, who afterwards made similar experiments, used methods similar to those of Coulomb.

Allusion has already been made to the use of the proofplane, and it has been stated that when applied to any part of the surface of an electrified body, it brings away just as much electricity as originally occupied the part of the surface which it covers. If, therefore, we electrify the movable ball of the torsion balance in the same sense as the body we are to examine, and note the repulsion caused by the proof-plane when introduced in place of the fixed ball after having touched in succession two parts of the surface of the body, we can, from the indications of the balance, calculate the ratio of the quantities of electricity on the plane in the two cases, and hence the ratio of the electrical densities at the two points of the surface. We suppose, of course, that the proof-plane is small enough to allow us to assume that the electrical density is sensibly uniform over the small area covered by it. In some of his experiments Riess used a small sphere (about two lines in diameter) instead of the small disc of the proof-plane as Coulomb used it. The sphere in such cases ought to be very small, and even then, except in the case of plane surfaces, its use is objectionable, unless the object be merely to determine, by twice touching the same point of the same conductor, the ratio of the whole charges on the conductor at two different times. The fundamental requisite is that the testing body shall, when applied, alter the form of the testing body as little as possible, and this requisite is best satisfied by a small disc, and the better the smaller the disc is. The theoretically correct procedure would be to have a small portion of the actual surface of the body movable. If we could remove such a piece so as to break contact with all neighbouring portions simultaneously, then we should, by testing the electrification of this in the balance, get a perfect measure of the mean electric surface density on the removed portion. We shall see that Coulomb did employ a method like this.

There are various ways of using the torsion balance in researches on distribution. We may either electrify the movable ball independently (as above described), or we may electrify it each time by contact with the proof-plane when it is inserted into the balance. It must be noticed that the repulsion of the movable ball is in the first case proportional to the charge on the proof-plane, but in the second to the square of the charge, so that the indications must be reduced differently.

In measuring we may either bring the movable ball to a fixed position, in which case the whole torsion required to keep it in this position is proportional to the charge on the proof-plane (or to its square, if the second of the above modes of operation be adopted), or we may simply observe the angle of equilibrium and calculate the quantity from that. It is supposed, for simplicity of explanation in all that follows, that the former of the two alternatives is adopted, and that the movable ball is always independently charged.

The gradual loss of electricity experienced more or less by every insulated conductor has already been alluded to. This loss forms one of the greatest difficulties to be encountered in such experiments as we are now describing. If we apply the proof-plane to a part of a conductor and take the balance reading, giving a torsion τ 1 say, and repeat the observation, after time Z, we shall get a different torsion τ 2 , owing to the loss of electricity in the interval. This loss, partly if not mainly due to the insulating supports, depends on a great many circumstances, some of which are entirely beyond even the observation of the experimenter. We may admit, however, what experiment confirms within certain small limits, that the rate of loss of electricity is proportional to the charge, and we shall call r - 1 - -- 2 (the loss per unit of time on hypothesis of uniformity) the coefficient of dissipation (δ). This coefficient, although, as we have implied, tolerably constant for one experiment, will vary very much from experiment to experiment, and from day to day; it depends above all on the weather.

Supposing we have determined this coefficient by such an observation as the above, then we can calculate the torsion τ', which we should have observed had we touched the body at any interval f after the first experiment; for we have, provided t! be small,

τ' = τ 1 - δf' = τ 2 + δ(i - f ).

In particular, if c = J t, we have

τ' = ⅜(τ 1 + τ 2 ).

Coulomb used this principle in comparing the electric densities at two points A and A' of the same conductor. He touched the two points a number of times in succession, first A, then A', then A again, and so on, observing the corresponding torsions τ 1 , τ 1 ', τ 2 , τ 2 ', ιkc., the intervals between the operations being very nearly equal. He thus got for the ratio of the densities at A and A' the values T1 _- Tg> 2τ 1 -^ 2 - -⅜-,-> <kc. These values ought to be all

τ 1 +τ 2 ' , 2t 2 ' equal: the mean of them was taken as the best result.

In certain cases, where the rapidity of the electric dissipation was too great to allow the above method to be applied, Riess used the method of paired proof-planes. For a description of this, and for some elaborate calculations on the subject of electrical dissipation, the reader is referred to Riess’s work.

The cage method is well adapted for experiments on distribution. The proof-plane, proof-sphere, or paired proof-planes may all be used in conjunction with it. If the cage be fairly well insualted, and a tolerably delicate Thomson’s electrometer be used, so that the cage may be made large, and the surface density on its outside therefore small, there will be little loss of the external charge; and the method has this advantage, that dissipation from the proof-plane inside the cage does not affect the result of the measurement in hand, it being indifferent, qua effect on the electrometer, whether the electricity inside the cage be on the proof-plane, in the air, or elsewhere, provided merely it be inside. The state of the cage as to electrified air, àc., is easily tested by the electrometer at any time.

Coulomb's Results.—If we electrify a sphere, and test the electrical density at two points of its surface, experiment will show, as would be expected from the symmetry of the body, that the density at the two points is the same. If we test the electric density at any point of a sphere, and then halve its charge by division with an equal neutral sphere, and test the electric density again, we shall find it half what it was before. The electric density at any point is therefore proportional to the whole charge on the sphere, or to the mean density, meaning by that the whole charge divided by the whole surface of the sphere.

If, instead of a sphere, we operate with an ellipsoid generated by the revolution of an ellipse about its major axis, we shall find that the electric density is not uniform as in the case of the sphere, but greater at the sharp ends of the major axis than at the equator, and the ratio of the densities increases indefinitely as we make the ellipsoid sharper and sharper. This leads us to state a principle of great importance in the theory of - electrical distribution, viz., that the electrical density is very great at any pointed part of a conductor.

If we determine the ratio of the densities at two points of an ellipsoid, diminish the charge, and redetermine the same ratio, we shall find that, although the actual densities are diminished, the ratio remains the same; and if we determine the density at any point of the ellipsoid, and then halve its charge b⅛ touching it with an equal and similar ellipsoid (they must be placed with their axes in the same straight line, and made to touch at the poles), and redetermine the density at the same point as before, we shall find that the density in the second case is half that in the first. We have in fact, in general, the important proposition that—

The density at any point of a conductor is proportional to the whole charge on the conductor, or, what is the same, to the mean density.

The following case given by Coulomb is interesting; it shows the tendency of electricity towards the projecting parts, ends, or points of bodies. The conductor was a cylinder with hemispherical ends,—the length of the cylinder being 30 inches, its diameter 2 inches. Coulomb gives the following results :— [table]

The density at the end is thus more than twice that at the middle.

Other results, taken from Coulomb’s unpublished papers, may be found in Biot, Mascart, or Riess. His results for a circular disc we shall quote further on.

Riess made a series of experiments on cubes, cones, <tc.; but as these are not of theoretical interest, the calculation in such cases being beyond the powers of analysis at present, and as the use of the proof-plane or sphere with bodies where edges and points occur is not free from objection, we content ourselves with referring to Riess’s work for an account of the results.

Coulomb made a series of experiments on bodies of different forms, which he built up out of spheres of different sizes, or out of spheres and cylinders. These are of very great interest, partly on account of the close agreement of some of the results with the deductions subsequently made by Poisson from the mathematical theory, and partly on account of the clearness with which they convey to the mind the general principles of electric distribution. His method in most cases was to build up the conductor and electrify it with all the different parts in contact, and then after separating the parts widely, to determine the mean density or the whole amount of electricity on each part by the proof-plane or otherwise.

For spheres in contact he found the following results,— S, Q, σ; S', Q', σ' denoting the surface, quantity of electricity, and mean surface density for the two spheres respectively. [table]

From this it appears that although the whole amount of electricity on the large sphere is greater than that on the small, yet the mean density for the smaller sphere is greater than for the larger. The above result also affords an experimental illustration of the action of the earth in discharging a conductor connected with it. Comparing the conductor to the small sphere and the earth to the large sphere of 62 times the superficial area of the small one, if we start with charge Q on small sphere and then put the two in contact, the charge on the small sphere will be reduced to Q, so that the mean density is dimin- 3o*b ished in the ratio 1: 38 ’ 6. This ratio increases indefinitely S'

as the ratio ‘4 increases. These results are in satisfactory agreement with Poisson’s calculations. Coulomb was led by his observations to assign 2 as the limit of the ratio of the mean densities when the ratio of the diameters of the spheres is infinitely great; the mathematical theory gives ⅞or1∙65.

υ

Coulomb also determined the density at the apex or smaller end of the body formed by two unequal spheres in contact. The following are his results, the mean density of the larger sphere being unity :— [table]

When two equal spheres are placed in contact the distribution will of course be the same in each; Coulomb found that, from the point of contact up to a point on the surface of either sphere distant from it by about 20°, no trace of electricity could be observed; at 30°, 60°, 90’, 180° respectively, the electric density had the relative values ‘20, '77, , 96, 1Ό0. When the spheres are unequal the distribution is no longer alike on each. On the small sphere it is less uniform, and the density at the point of the small sphere diametrically opposite the point of contact is greater than anywhere else on the body. The distribution on the larger sphere is more uniform than on the smaller, and the more unequal the spheres are the more uniform is the distribution on the larger, and the smaller the unelectrified part in the neighbourhood of the point of contact.

The following results of Coulomb are useful illustrations of distribution on elongated and .pointed bodies :—

Three equal spheres (2 in. diameter) in contact, with their centres in the same straight line: the mean densities were 1 '34, 1Ό0, 1'34 on the spheres 1, 2, and 3 respectively.

Six equal spheres as before: mean densities on 1, 2, and 3 = 1∙56, l∙05, l∙00.

Twelve equal spheres: mean densities on 1, 2, and 6 = 1 ’ 7O, 1Ί4, l∙00.

Twenty-four equal spheres: mean densities on 1, 2, and 12 = Γ75, l∙07, l∙00.

Large (8 in. diameter) sphere with four small (2 in.) spheres applied to it, all the centres in line: the mean density on large sphere being 1, that on the small one next it was ’ 60 that on the extreme small one 2 - 08.

Large sphere 1, and twenty-four (2 to 25) small ones: mean densities on 1, 2, 13, 24, 25 = ΓOO, ’ 60. Γ28, 1'46, 2T7.

MATHEMATICAL THEORY OF ELECTRICAL EQUILIBRIUM.

We take as the basis of our theory the assumptions already laid down under the head Provisional Theory, and in addition the precise elementary law of electrical action established by Coulomb. We shall also suppose that we have only perfect conductors and perfect non-conductors to deal with, the medium being in all cases the same, viz., air. When we have to deal with electrified non-conductors we shall suppose the electrification to be rigid, i.e. incapable of disturbance by any electric force we have to consider.

In our mathematical outline we have in view the requirements of the physical more than the mathematical student, and shall pass over many points of great interest and importance to the latter, for full treatment of which we must refer him to original sources, such as the classical papers of Green, the papers of Sir William Thomson, and the works of Gauss. Of peculiar interest mathematically is the elegant and powerful memoir of the last—Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs-und Abstossungskräfte, in which will be found detailed discussions of the continuity of the integrals used in the potential theory, &c. The works of Green and Thomson are too well known in this country to require farther remark.

When, in what follows, we speak of the electric field, we mean simply a portion of space which we are considering with reference to its electrical properties; it will be found conducive to clearness to regard that space as bounded. In general the natural boundary would be the walls of the experimenting room; but, for mathematical purposes, we shall, unless the contrary is stated, suppose our field to be bounded by a sphere of radius so great that the action at a point on its circumference due to an electrified body in the field is infinitely small.

The resultant force at a point in the electric field is the force which would be exerted on a unit of + electricity placed there without disturbing the electrical distribution elsewhere. It is plain that the resultant force has a definite magnitude and direction at every point in the field, and consequently is in modern mathematical language a vector. A curve drawn in the field such that its tangent at every point is in the direction of the resultant force at that point is called a line of force. We can draw such a line through every point of space, and if we suspend at any point a small conducting needle, it is obvious, from what we have already laid down about induction, that it will take up a position very nearly parallel to the line of force; so that if we start from any point and carry the centre of the needle always in the direction in which the needle points we should trace out a line of force.

The potential at any point is the work done by a unit of + electricity in passing from that point to the infinitely distant boundary of the electric field, the electric distribution being supposed undisturbed. It is usual to call the infinitely distant boundary a place of zero potential. Zero is to be understood in the sense of “point or position from which we reckon.”

Consider two points P, Q, infinitely near each other in the field, and draw a curve from P passing through Q to ». Then, if F be the component parallel to PQ of the resultant force at P, we have by our definition F.PQ = V p -V q j or in differential notation

F<Z⅛ ’ = -dV, hence F=-^X (1), as and V=∕*Fds=∕"(Xrfz + Ydy + Z√2) . . . (2), where V denotes the potential at P, and X,Y,Z the components parallel to the co-ordinate axes of the resultant electric force. We clearly have as particular cases of (1)

x σ Y _ z _ ≤

X ~~dx Ύ ~~ dy dz · w *

We may remark that, in all cases which we shall consider at present, the work done in passing from any point to any other point is the same whatever the intermediate path of our exploring unit. Hence V as above defined is a single valued function, and th' formulae (3) gives the components of resultant force when V is known.

The work done by a unit of + electricity in passing by any path from P to Q is called the electromotive force from P to Q; it is obviously equal to the difference of the potentials at the two points. Thus v p- v Q√^(x ^+ γ ⅜ + iW2) · · · (4). is the electromotive force from P to Q.

Suppose we concentrate m units of electricity at any point P, and require the potential due to this at a point Q, distant D from l ’. Applying (2), and, since any path to ∞ may be chosen, taking the integral along the production of PQ to ∞, we get

∕" 0 °

∙tr ∕ Z7t j Trt e~

n ~∕ ~ι dr ~V i . . . . (5).

.∕d r i D

If we have any number of discrete points with charges r>ι 1 , rn i , m 3 ,.. .. at distances D 1 , D 2 , D 3 , from Q, since the work done by the exploring unit under the action of the whole is got by adding up the work done under the action of each part separately, we clearly have

,. w, rn., m v = ⅛ + i⅛ + icc ∙= 2 D · · · · (6)∙ From this we may pass to thecase of a continuous volume distribution. If p be the volume density at the point ξηζ, and V the potential at xyz, we have v =^7^<⅛≤ (7 ), where D denotes + J(ξ- χ ) 1 +(n-yf+(ζ-^) i , and the integral is to be extended over every part of the field where there is any charge,—or, which is the same thing, over the whole field, on the understanding that p = 0 where there is no charge.

If, as will generally be the case, the electricity is distributed on a surface in such a way that on an element dS of surface there is a quantity <rdS of electricity, where σ is a finite surface density, then N ‘ffτr ' <S) ·

where D has the same meaning as before, and the integral is extended all over the electrified surface or surfaces.

We may make here the important remark that, so long as p or <r is not infinite, the integrals in (7) and (S) are finite and continuous. This depends on the fact, which we cannot stop to prove, that the part of tne potential at I’, contributed by an infinitely small portion of electricity surrounding P, is infinitely small.

In practice, therefore, the electric potential is always continuous; for although we may in theory speak of discrete points and electrified lines where finite electrification is condensed into infinitely small space, yet no such cases ever occur in nature. It may also be shown for any electrical system of finite extent, that, as the distance of P from 0, any fixed point at a finite distance from the system is increased indefinitely, the potential at P approaches more and more nearly the value , where M is the algebraical sum of all the electricity in the system, and D the distance of P from 0. Hence at any point infinitely distant from Ο, V = 0.

We next proceed to prove the following proposition, which will form the basis of the subsequent theory :—

The surface integral of electric induction taken all over the surface inclosing any space is equal to 47γ times the algebraical sum of all the electricity in that space.

By the electric induction across any element of the surface (taken so small that the resultant force at every point of it may be regarded as uniform) is meant the product of the area of the element into the component of the resultant force in the direction of the normal to the element which is drawn outwards with respect to the inclosed space. Thus JS being an element of surface, ∈ the angle between the positive direction of the resultaut force R and the outward normal ν, and E the sum of all the electricity in the inclosed space, the proposition in symbols is— #Rcos€<7S= 4τrE . . . . (9).

We shall prove it in the manner most naturally suggested by the theory of electrical elements acting at a distance, by first showing that it is true for a single element e either outside or inside the surface. Let us suppose e to be at a point P, fig. 11, within S, which for greater generality we may suppose to be a re-entrant surface. Draw a small cone of vertical solid angle ιl< 1 > at P, and let it cut the surface in the elements QR, Q'R', Q''R"; let the outward normals to these be QM, Q'M', Q''M". The elements of the surface integral contributed by QR, Q'R ’, Q"R" are obviously QR« cos e rfωPR 2

PR» , &c.; but QL - cose ,

<7ωPR" j 1 i - t <,p. - dwPli" 2

QR = — 3 f r > aπα *∙<! κ — cos f >∣; hence the three elements of the integral become +edω,-eclω, +edω; and the sum is edω. Adding now the contributions from all the little cones which fill up the solid angle of 4π about P, we get

Jjll cos tdS = ejTdω = 4ιτe.

Had the point P been outside, the numbers of emergences and entrances would have been equal, the contribution of each cone zero, and on the whole

√7R cos∈i7S = 0.

Combining these results, we see that the proposition is true for a single element. Hence, by summation for all the elements, we can at once extend it to any electrical system; for all the elements external to S give zero, and all the internal elements give 4-π∙2e = 4π∙E.

Let us apply the above proposition to the space enclosed by the infinitely small parallelepiped whose centre is at xyz, and the co-ordinates of whose jyigles are x±⅛dx, y±,∖dy, z±⅛dz. The contributions to the surface integral from the two faces perpendicular to the «-axis «re -(g + and (g - *L g)⅛*,

Adding these and the four parts from the remaining sides, and equating to iτrpdxdydz, which is the 4πE in this case, we have d ’ V . d≡V , d»V .

<⅛> + ⅜≡- + *γ + 4 ' ,, = 0 · or, is it is usually abbreviated,

V 2 V + 4r p = (10).

Equation (10), originally found by Laplace for the casep — 0, and extended by Poisson, has been called the characteristic equation of the potential. It may be applied at any point where ρ is finite and the electric force continuous. It might be shown by examining the inte-dV

grals representing X,Y,Z,and— , &c., that the electric force is con-dx tinuous wherever there is finite volume density. Equation (10) may be looked on either as an equation to determine the potential when p is given, or as an equation to determine p whenV is given. We shall have occasion to use it in both ways.

The characteristic equation cannot be applied in the form (10) when the resultant force is discontinuous. This will be found to be the case at a surface on which electricity is distributed with finite surface density. Let us consider the values of the resultant force at two points, P and Q, infinitely near each other, but on opposite sides of such a surface. Resolve the resultant force tangentially and normally to the surface. If we consider the part of the force which arises from an infinitely small circular disc, whose radius, though infinitely small, is yet infinitely great compared with the distance between P and Q, we see that infinitely little is contributed to the tangential component at P or Q by this disc, while it can be readily shown that the part of the normal component arising therefrom is 2τrσ, directed from the disc in each case, when σ is the surface density. Hence, since the part of the resultant force arising from all the rest of the electrified system obviously is not discontinuous between P and Q, the tangential component is continuous when we pass through an electrified surface, but the normal component is suddenly altered by 4πσ.

For a thorough investigation of these points the reader is referred to Gauss or Green. We can arrive very readily at the amount of the discontinuity of the normal force by applying (9) to the cylinder formed by carrying an infinitely short generating line round the element dS, so that one end of the cylinder is on one side of dS and the other on the other, the lateral dimensions being infinitely small, but still infinitely greater than the longitudinal. The only part of the integral which is of the order of dS is the part arising from the two ends; hence if N,N' be the value of the normal components on the two sides of S, we clearly get

(N - N') <ZS = 4π<rdS , or N - N ’ = 4π<τ.

If V 1 , V 2 denote the potentials on the two sides of S, and v l , t> 3 the normals to dS, drawn towards these sides respectively, then we may obviously write our equation dv.dv i ι

⅜ i + σ = ° (U) ·

V ritten in this form the equation has been called the surface characteristic equation of the potential. It may be looked upon as a surface condition, which must be fulfilled by the values of V on the two sides of an electrified surface on which the surface density σ ’ is given, and where, in consequence, there is discontinuity in the first differential coefficients of λ; or it may be looked on as an equation to determine σ when A , and V 2 are given.

M e have seen that we can draw through every point of the electric field a line of force. A surface constructed so that the potential at every* point of it has the same value is called an equipotential or level surface. We can obviously draw such a surface passing through every point of the field. It is clear, too, that the line of force at every point must be perpendicular to the level surface passing through that point, lor since no work is done on a unit of + electricity in passing from one point of a level surface to a neighbouring point, there can be no component of the resultant force tangential to the surface; in other words, the direction of the resultant force is perpendicular to the surface. This is expressed otherwise by saying that the lines of force are orthogonal trajectories to the level surface.

If we take a small portion of a level suιface, and draw through every point of the boundary a line of force, we shall thus generate a tubular surface which will cut orthogonally every level surface which it meets. Such a surface is called a tube of force.

Let a tube of force cut two level surfaces in the elements <ZS and cZS'. Apply to the space contained by the part of the tube between the surfaces our fundamental equation (9). We thus get, since there is no normal component perpendicular to the generating lines of the tube,

R<7S-R'<7S' = 0, .... (12), provided the tube doe3 not cut through electrified matter between the two surfaces. Here R and R' denote the resultant force at √S and <7'S, which are supposed so small that the force may be considered uniform all over each of them. It appears then that the product of the resultant force into the area of the normal section of a tube of force is constant for the same tube so long as it does not cut through electrified matter; or what amounts to the same, the resultant force at any point of a tube of force varies inversely as the normal section of the tube at that point.

If we divide up any level surface into a series of small elements, such that the product R<7S is constant for each element and equal to unity, and draw tubes of force through each small element, then the electric induction through any finite area of the surface is equal to the number of tubes of force which pass through that area; for ij n be that number, we have, summing over the whole of the area— SR(7S = n (13), the left band side of which is the electric induction through the finite area. It is clear, from the constancy of the product Rd’S for each tube of force, that if this is true for one level surface it will be true for every other cut by the tubes of force. It is evident that the proposition is true for any surface, whether a level surface or not, as may be seen by projecting the area considered by lines of force on a level surface, and applying to the cylinder thus formed the surface integral of electric induction, it being remarked as obvious that the same number of tubes of force pass through the area as through the projection. This enables us to state the proposition involved in equation (9) in the following manner :—

The excess of the number of tubes of forces whichleανe a closed surface over the number which enter is equal to 4τr times the algebraical sum of all the electricity within the surface.

{N.B.—The positive direction of a line of force is that direction in which a unit of + electricity would tend to move along it.) This proposition enables us to measure the charge of a body by means of the lines of ⅛rce. We have only to draw a surface inclosing the body, and very near to it, and count the lines of force entering and leaving the surface. If the number of the latter, diminished by the number of the former, be divided by 4τr, the result is the charge on the body.

If we apply ( 13) to a portion of an equipotential surface so small that R may be considered uniform over the whole of it, we may write r -⅛ or in words :—The resultant force at any point is equal to 1 the number of lines of force per unit of area of level sτerface at that point, meaning thereby the number of lines of force which would pass through a unit of area of level surface if the force were uniform throughout, and equal to its value at the point considered.

We are now able to express by means of the lines of force the resultant force at any point of the field, and the charge in any element of space. The electrical language thus constructed was invented by Faraday, who continually used it in his electrical researches. In the hands of Sir William Thomson, and particularly of Professor Clerk Maxwell, this language has become capable of representing, not only qualitatively, but also quantitatively, with mathematical accuracy, the state of the electric field. It has the additional advantages of being well fitted for the use of the practical electrician, and of lending itself very readily to graphical representation.

It will be convenient, before passing to electrical applications, to state here another general property of the potential which follows from our fundamental proposition.

The potential cannot have a maximum or minimum value at a point where there is no electricity.

For if a maximum value were possible, we could draw round the point a surface at every point of which the potential was decreasing outwards; consequently at every point of this surface the normal component of the resultant force in the outward direction would be positive, and a positive number of lines of force would leave the surface. But this is impossible, since, by our hypothesis, there is no electricity within. Similarly there could be no minimum value.

From this it follows at once that if the potential have the same value at every point of the boundary of a space in which there is no electrified body, then the potential is constant throughout that space, and equal to the value at the boundary. For if the potential at any point within had any value greater or less than the value at the boundary, this would be a case of maximum or minimum potential at a point in free space, which we have seen to be impossible.

In order that there may be electrical equilibrium in a perfect conductor, it is necessary that the resultant electric force should be zero at every point of its substance. For if it were not so at any point the positive electricity there would move in the direction of the resultant force and the negative electricity in the opposite direction, which is inconsistent with our supposition of equilibrium. This condition must be satisfied at any point of the conductor, however near the surface. At the surface there must be no tangential component of resultant force, otherwise electricity would move along the surface. In other words, the resultant force at the surface must be normal; its magnitude is not otherwise restricted; for by our hypothesis electricity cannot penetrate into the non-conducting medium.

These conditions are clearly sufficient. We may sum them up in the following single statement :—

If the electricity in any conductor is in equilibrium, the potential must have the same value at every point in its substance.

For if the potential be constant, its differential coefficients are zero, so that inside the conductor the resultant force vanishes. Also the surface of the conductor is a level surface, and therefore the resultant force is everywhere normal to it. This constant value of the potential we shall henceforth speak of as the potential of the conductor.

Since the potential is constant at every point in the substance of a charged conductor, we have at every point V 2 V = 0, and hence by the equation of Poisson p = 0; that is, there is no electricity in the substance of the conductor. We thus get, as a theoretical conclusion from our hypothesis, the result already suggested by experiment, that electricity resides wholly on the surface of conductors.

If we apply the surface characteristic equation to any point of the surface of a conductor, we get 1 dy - r ∕ik∖ σ =~Τ. d~ ν ~Γτ> · · · < 15 λ

which gives the surface density in terms of the resultant force and reciprocally.

We may put this into the language of the lines of force by saying that the charge on any portion of the surface of a conductor is equal to the number of lines of force issuing from it divided by 4τr.

Since the surface of a conductor in electric equilibrium is always a level surface, it follows, from what we have already proved about a space bounded by a surface of constant potential, that, inside a hollow conductor the potential is constant, provided there be no electrified bodies within. This is true, no matter how we electrify the conductor or what electrified bodies there may be outside. Hence, if we inclose any conductor A completely within another B, then electrify B and put A in metallic communication with it, A will not become charged either + or -; for, A being at the same potential as B, electricity will not tend to flow from the one to the other. This is in reality Biot’s experiment with the hemispheres, to which we have already alluded; only the point of view is slightly changed. The most striking experiment ever made in illustration of the present principle is that described by Faraday in his Experimental Researches. He constructed a hollow cube (12 feet in the edge) of conducting matter, and insulated it in the lecture-room of the Royal Institution. We quote in his own words the part of his description which bears on the present question :—

“1172. I put a delicate gold leaf electrometer within the cube, and then charged the whole by an outside communication, very strongly for some time together; but neither during the charge or after the discharge did the electrometer or air within show the least sign of electricity 1 went into the cube and lived in it, and using all other tests of electrical states, I could not find the least influence upon them, though all the time the outside of the cube was powerfully charged, and large sparks and brushes were darting otι from every point of its outer surface.”

The proposition that the potential is constant inside a hollow conductor containing no electrified bodies may be regarded as one of the most firmly established in the whole of experimental science. The experiments on which it rests are of extreme delicacy. It is of the greatest theoretical importance; for we can deduce from it the law of the inverse square. Taking the particular case of a spherical shell, uninfluenced by other bodies, on which of course the electrical distribution must from symmetry be uniform, it can be demonstrated mathematically that, if we assume the action between two elements of electricity to be a function of the distance between them, then that function must be the inverse square, in order that the potential may be constant throughout the interior. A demonstration of this proposition was given by Cavendish, who saw its importance; a more elaborate proof λvas afterwards given by Laplace; for a very elegant and simple demonstration we refer the mathematical reader to Clerk Maxwell’s Electricity, vol. i. § 74. This must be regarded as by far the most satisfactory evidence for the law of the inverse square; for the delicacy of the tests involved infinitely surpasses that of the measurements made with the torsion balance; and now that we have instruments of greatly increased sensitiveness, like Thomson’s quadrant electrometer, the experimental evidence might be still further strengthened.

In the problem to determine the distribution of electricity in a given system of conductors, the data are in most cases either the charge or the potential for each conductor. If the conductor is insulated it can neither give nor lose electricity, its charge is therefore given. If, on the other hand, it be connected with some inexhaustible source of electricity at a constant potential, its potential is given. Such a source the earth is assumed to be; and we shall henceforth take the potential of the earth as zero, and reckon the potential of all other bodies with reference to it. If all our electrical experiments were con ducted in a space inclosed by a perfectly conducting envelope, the potential of this envelope would be the natural zero of our reckoning.

It will be useful to analyse more closely the distribution on a system of conductors, in order to see how far the above data really determine the solution of the electrical problem. For this purpose the following proposition is useful. If e 1 , e 2 , . . . . e n be the charges at the points 1, 2, . . . . n of any system, and V the potential at P, and if V' be the potential at P due to e 1 ', e 2 , . . . . c n ' at

1, 2, . . . . n, then the potential at P due to e 1 + e 1 ', e 2 + e 2 ', . . . . at 1, 2, . . . . is V + V'. This principle follows at once from the definition of the potential as a sum formed by the mere addition of parts due to all the single elements of the system.

Applied to a system of conductors in equilibrium, it may evidently be stated thus: If E 1 , E 2 , . . . . E n and V 1 , V 2 , . . . . V n be the respective charges and potentials for the conductors 1, 2, 3 . . . . n in a state of equilibrium, E 1 ', E 2 ', .... E n ' and V 1 ', V 2 ’ j .... V n ' corresponding charges and potentials for another state of equilibrium, then E 1 + E 1 '. . . . E„ ÷ E√, V 1 + V 1 ’. . . . V n ÷ V n ' will be corresponding charges and potentials for a third state of equilibrium.

Suppose that in the system of conductors 1, 2, 3 n the con ductor 1 is kept at potential 1 and all the others at potential zero, then it can be shown that there is one and only one distribution of electricity fulfilling these conditions. Mathematically stated, the problem is to determine a function V, which shall satisfy the equation v 2 V=0 throughout the space unoccupied by conductors, and have the values 1, 0, 0, 0 was respectively at each point of the surfaces of 1, 2, « respectively.

Consider the integral

τ ∕V∕*tdVΙ a ^<W∣ i j ,

∖dx∖ + df∖ + ~dz∖∖ dxdydz - * · * < 16 >'

where the integration is extended all over the space unoccupied by conductors. If we consider all the values which this integral may have, consistent with the boundary conditions V = l, V = 0,

&c. at the surfaces of 1, 2, &c., it is obvious that there must be a minimum value; for the integral is essentially positive, and cannot become less than zero.

Now δl = 2 ‰ kz ∙ } ixdydz

= 2Wv 2 Vdxdydz (17)

by partial integration. The surface terms vanish, since δ V = 0 at every surface. Hence v 2 V∙=0 is the condition for a maximum or minimum value of I, and since we know that a minimum value exists, there must be a solution of this equation. It can, moreover, be shown, by a method which we shall apply below to the more general problem, that there is only one solution of v 2 V = 0 consistent with the given conditions, and this will of course be that which makes I a minimum. If our mathematical methods were powerful enough to determine V, we might proceed to find the surface density for each conductor by means of the formula 1 dV

<r =; then the charges on the conductors could be found by means of the integral - J^J^‘ ver T ^ ew cases indeed could we actually find these cnarges; we have, however, demonstrated their existence and shown that our problem is definite.

Let these charges on 1, 2,.. . n be called p 9ι 2 · · · «·

Corresponding to the data 0, 1, 0, . . . . 0 for the potentials of 1, 2,.. . n, we should get a series of charges q 2 v q 22 ,.... q 2 n , and so on; q 11 , q 2 2 , q 3 3 ... are called the coefficients of self-induction or capacity for the conductors 1, 2, 3, . . .; q l 2 , q l 3 , <tc., are called the coefficients of induction of 1 on

2, 1 on 3, Ac. It is obvious that these coefficients depend solely on the form and relative position of the conductors. It follows, from the principle of the superposition, that, if 1, 2, . . . n be at the potentials V 1 , 0, 0 . . . 0, then the çharges on them will be 7 1 l V 1 , 7 12 V 1 .... 2- „V V e may construct then a series of states of equilibrium represented thus :— [table]

and so on. Superposing all these, we get a system in equilibrium, in which the potentials are V 1 , V 2 , . . . V n , and the charges

E 1 =iιιV 1 + g al V s+ .. . ⅛5n 1 V n 1

E2 = 7ι Wι + g ’ a2 V 8 + . . . + q ni V n ?... (18). &c. =&c. )

It appears therefore that the 2n quantities E 1 , àc., V 1 , &c., are connected by n linear equations; so that when n of them are given, the rest can be determined in terms of these in a definite manner.

Returning then to our general problem, we see that, when either the charge or the potential is given for each conductor, the electrical problem is determinate, and a solution is given by the linear equations of (18). The potential at any point of the field can be written down very easily. Suppose in fact v 1 to be the value at the point P of the function V which we determined in solving the case where the potentials 1, 0, 0,. . . . 0 are given for 1, 2, ... n, v 2 the corresponding function for the case 0, 1, 0, . . . 0, and so on. Then the potential at P in the general case is obviously V = V λ + V 2 v 2 + . . . .+V n r n . . (19), where v 1 , v 2 , , . . v n are all known functions, and V 1 , V 2 , . . . V n are all either given, or determined in terms of given quantities by the equations (18).

It is very easy to show that there is no other solution of the problem than the one we have found.

Suppose in fact that V' is a function different from V, which satisfies all the conditions of the problem. Consider the function U = V-V', since V and V' both satisfy the equation v 2 V = 0 , we have v 2 U-0. Also at surfaces where V is given U = 0. At surfaces where V is not given, we have U = constant - constant = constant; and, since in this case the charge wiΓ be given, we shall have f∕dν dS =ffS^ ds ! and therefore j ∕∕* ~ dS = 0.

Now we have

∕ZZ i (S^ s-)' +4c ∙ i ,ω ∙ ,, ≈ ∕z∕isr÷^÷⅛r^∙∙ ~fflT v UdS ~jy∕ j v 2 Udxdydz.

The first term vanishes for all the surfaces,—for some because U ≈ 0, for others because U is constant and; and the second term vanishes because v 2 U = 0.

Hence the integral on the left hand must vanish, and that too element by element, since every element is positive. Hence we must have dV dV ’ dy dV ’ dV dV' dx dx ’ dy ~ dy dz ~~dz ’

Hence V and V' can only differ by a constant. But such difference is precluded by the boundary conditions. Hence the functions are identical; in other words, there is but one solution to the problem we have proposed.

It is very easy to show, by methods of which we have already had an example, that the value of V thus found makes the integral

⅛∕∕∕Gf÷I÷fi>" a minimum. Now, we shall show directly that this inte gral represents the potential energy of the system. It follows, therefore, that the distribution which we have found is in stable equilibrium.

If we solve the equations (18), we shall get v i=-Pll e 1+2⅛1 e 2 + . . . . Pnι En ) V,=Λ 2 E 1 +p a 2 Ej+ . . . . pn i E n 1 (20).

&c. )

A set of equations which we might obviously have arrived at by first principles. The physical meaning of the coefficients ρ 11 , p χ 2 , and p x n is very obvious; they are the potentials, corresponding to a state of equilibrium, in which the charges on I, 2, 3,.... n are 1, 0, 0, ... 0, and so on. p χ 1 , p 12 ,.. &c., are called coefficients of potential; and, mutatis mutandis, all the remarks already made about ?! V ?i 2’ &c., apply to them. Many interesting and important theorems have been proved about these coefficients, for which we refer the reader to Maxwell {Electricity, vol. i. chap. 2), whose treatment of the subject we have in the main been.following. One of these, of great importance, we shall prove here, because it leads us to state a very important general theorem, which we shall have occasion to use again.

The mutual potential energy of two electrical systems, A and B, is the work done in removing the two systems to an infinite distance from each other, the internal arrangement of each system being supposed unaltered during the process. It is clear that we may suppose either that A is fixed and B moves off to infinity, or that B is fixed and A moves; the work done in both cases is, by Newton’s third law of motion, the same. This is sometimes expressed by saying that the potential of A on B is the same as that of B on A.

In fact, the expression for the mutual potential energy is 2⅞ (21), where q is any element of electricity belonging to A, and q ’ any element belonging to B, and D is the distance between them, the summation being extended so as to include every pair of elements. We may arrange (21 ) as follows :—

Jl' 2 l⅞ +2i' 2 -⅜ + fec ∙.

each group belonging to a point in B, or, as we may write it, g√V 1 + y 2 ' V i -f-⅛c., or 2√V.

We may also arrange (21) in the form 2 1 ^+7 i 2,^ + ⅛c., each group belonging to a point in A. Hence we have the following equalities: —

2<Z ’ V = 2^' = 2 , 7 V' (22).

The first and last of these expressions are called respectively the potential of A on B, and the potential of B on A, and this equality explains the statement made above.

The two systems A and B may be different states of equilibrium of the same system, if we choose. In this case we may still farther modify the expression in (22), and write

V 1 ¾'+W+te=V⅛V,⅝ 1 +Λc. (See Gauss, Le.)

So that we may state the proposition thus :—If E 1 , E 2 , .. . E n , V 1 , V 2 ,.. . V n , and E 1 ', E 2 ',... E n , V/, V 2 ',. .. V n ' be the respective charges and potentials of the conductors in two different states of equilibrium, then we have

SE'V = SEV' .... (23).

If we take for the two states of the system [table]

equation (23) becomes

2ii = 2i2 · · · · (24), or, in words, the coefficient of induction of I on 2 is equal to that of 2 on 1.

There is one more geueral theorem on electrical distribution which, from its great practical importance, deserves a place here. Suppose we take a hollow conductor of any form, place any electrical system inside it, and connect the conductor with the earth, then equilibrium will be established, in such a way that the potential of every portion of the conductor is zero. Now, the potential being zero at all infinitely distant points, we may regard the outside space as inclosed by a surface of zero potential; hence the potential at every point in this space must be the same, and there can be no electrical action anywhere outside.

Again, removing the internal system, let us place any system outside the conductor, and, besides, charge it to any desired extent, keeping it insulated this time. Then the outer and inner surfaces of the conductor will be level surfaces; aud, since there is no electricity inside the inner surface, the potential in the interior will be constant. Hence the external.system, in a state of equilibrium, exerts no action whatever within. Now we may evidently, without mutual disturbance, superpose such an internal and external system as we have described, and still get a system in equilibrium. It is, moreover, clear that we can in this way satisfy the most general conditions that can be assigned. Hence, since we know that there can be only one solution of the problem of electrical equilibrium, the synthetical one thus obtained represents the actual state of affairs. When, therefore, a hollow conductor with any external and internal systems is in equilibrium, the equilibrium of the internal is independent of that of the external system.

Moreover, if we draw any surface ⅛ the substance of the hollow conductor, no lines of force cross it in one direction or the other; therefore the whole amount of electricity within must be zero; in other words, the charge on the internal surface of the conductor is equal and opposite to the algebraical sum of the charges on all the bodies ιvithin.

. These propositions contain the principle of what are called electrical screens, i.e. sheets of metal used to defend electrical instruments, <fcc., from external influences. On the practical efficiency of gratings in this way, see Maxwell (§ 203); on the application to the theory of lightning conductors, see a paper by him in the reports of the British Association for 1876.

If we take the simple case where there is no external system, but only a charge on the hollow conductor, we get a complete explanation of Faraday’s ice-pail experiment.

The potential energy of a system of charged conductors is the work required to bring them from a neutral state to the charges and potentials which they have at any time. The state of zero potential energy here contemplated is of course that in which there is an equal amount of + and - electricity everywhere in the system, or, as we might put it, the state in which there is no electrical separation. Now if Q denote the potential energy of the system, we have with the notation of (21)

Q-⅛ (25), the summation including every pair of elements in the system. If the system be in equilibrium, then, reasoning as above, it is obvious that SEV is just twice , inasmuch as each pair of elements will come in twice. Hence we get

Q = ⅛2EV (26).

This is an expression of the greatest importance. We can give it various forms; by means of (18) and (20) we get

Q = 4 r Γ T 2r ,V r V.= ⅛ 2 2∖,E r E, . . (27). r=l β=l r=l f=l

So that Q is a homogeneous quadratic function of the potentials or of the charges. If, therefore, we increase the potentials of all the conductors, or the charges of all the conductors in any ratio, we increase thereby the potential energy in the duplicate of that ratio.

We can by a transformation, which is a particular case of a theorem of Green’s, obtain a very remarkable volume integral for the potential energy of an electrical system.

Let V denote the potential at any point in the field. Consider the integral

<wezr<+>*∙ where the integration is to be extended throughout the whole of the space unoccupied by conductors. We have by partial integration and two similar equations. Hence where the surface integration extends’ over the surface of all the conductors, and it is to be noticed that dv is drawn from the conductor into the insulating medium. If ρ and σ be volume and surface densities, 1 dV , 1

σ -^4, and '--⅛ v v ∙

Thus we get

= ⅜J^J'Tσd ’ S + ∖J~J^J^N pdxdydz . . (28).

This result includes a more general case than our present one; for it shows that the potential energy of an electrical system is given by the integral on the left hand side in all cases, whether there is equilibrium or not. It is not even restricted to the case of perfect conductors and perfect nonconductors, for a slight modification of our preliminary statements would include that case as well. At present, however, we have p — 0 everywhere, and V constant at the surface and in the substance of each conductor, so that the right hand side is simply the expression ∣SEV which w e have already found for the potential energy; we may therefore write

< 29 >∙

R being the resultant force at any point of the field, and dv the element of volume. It is clear that we may if we like extend the integration over the whole field, since in the substance of any conductor R = 0.

When we know the potential energy of an electrical system it is very easy to find the force which resists or tends to produce any change of configuration. Two particular cases are of common occurrence and of considerable interest. First, let the charges on all the conductors be kept constant. Let the variable which is altered by the supposed change of configuration be <∕>, and let Φ be the correspond ing force tending to increase φ. Then, since no energy is supplied from without, if we suppose the displacemeut made infinitely slowly, so that no kinetic energy is generated, we have Φδ⅛> + δQ = O (30),

” ∙>=-⅛ < sl >∙

Referring to the second of the expressions in (27), we see that this may be written r=n ·—n Φ=-J2 2 E r E,⅛~ ·

r=l «=1 “Φ

From this it is evident that in similarly electrified states of the same system the force tending to produce a given displacement varies as the square of the electrification. It is important to remark that in the present case the system tends to move so that its potential energy is decreased.

Secondly, let us suppose that the potentials of the different conductors are kept constant during any displacement, energy being supplied from without.

We shall suppose the change made in two steps. First, we shall suppose the given displacement to take place while the charges remain constant. On this supposition the force exerted will,.to the first order of small quantities, be the same as that exerted when we suppose the potential not to vary; hence

Φδ≠ + ⅛2EδV = 0.

Next, supply energy from without so that the potentials become again V 1 , V 2 ,⅛c., . . . and the charges E 1 + δE 1 , E 2 + δE 2 , &c. The final result will be the same, to first order of small quantities, as if the two changes had been made simultaneously. Now, applying the theorem of mutual potential energy to the two states of our system,

Ell E 1 I E 2 I , EΙE 1 + δE 1 E 2 + δE 2 ∣

V ∣∣v∖+δτJ∖∖+δV^J · · · and vhrl^V I · · · ’ we have 2(E 1 + δE 1 )(V 1 + δV 1 )-2(EV), hence 2EδV=-2VδE (32);

, . 1 τ , dV 1 ττ dE dQ , therefore = ⅛≡V ^(V const.) . . (33).

By (27) this may be written r—n β=n d∩ .

Φ = I 2 2 V r v⅛ . r-1 f≡-l α<∕>

The energy supplied from without is

4{2(E + δE)V-∑E(V + δV) }

= ⅛2δEV- ⅛2EδV= - 2EδV = 2Φδ<∕> = 2δQ, by (32).

In other words, when the potentials of a system are kept constant by supply of energy from without, the system tends to move so as to increase the potential energy of electrical separation, and the amount of energy supplied from without is double this increase. If we suspend side by side two balls, each connected with the positive pole of a battery, the other pole of which is connected with the ground, the balls will tend to separate, and in separating they will gain with reference to gravity a certain amount δQ of potential energy; the charges on the balls will also increase to an extent representing an increase of electrical potential energy δQ, and the batteries will be drawn upon for an amount of 2δQ.

The problem of electrical equilibrium has been completely solved in very few cases. We proceed to give a short sketch of what has been done in this way, which may indicate to the reader what is known on this head.

We can deduce the distribution and potential in the case of an ellipsoid from known propositions about the attractions of ellipsoidal shells of gravitating matter.

Consider an ellipsoidal shell, the axes of whose bounding surfaces are (a, b, e){α+dα,b + db, c + de), where — = ⅞ = — = μ , The a b c potential of such a shell at any internal point is constant, and the equipotential surfaces for external space are ellipsoids confocal with (a, b, c). (See Thomson and Tait, §§ 519 sgq.) Hence if we distribute electricity on an ellipsoid (a, b, c) such that its density at every point is proportional to the thickness of the shell formed by the similar ellipsoids (a,b,c) (a + da,b + db,e + dc), the distribution will be in equilibrium. Thus if <τ = A0ρ, where θ is the thickness at any point and p the volume density of the shell; then the quantity of electricity on any element dS is A times the mass of the corresponding element of the shell; and if Q be the whole quantity of electricity on the ellipsoid, Q = A times the whole mass of the shell.

The mass of the shell is ∣πpd(α5c) = 4πμabcp, therefore Q = A4πμabcp. Also θ=μp where p is the perpendicular from the centre of the ellipsoid on the tangent plane. Whence we get

'-4⅛ < 34 >∙

that is, the density at any point varies directly as the distance of the tangent plane at that point from the centre.

Returning again to our ellipsoidal shell, we know that the resultant force at any external point P due to this shell is to that due to a “confocal shell” passing through the point in the ratio of the masses. Let the volume density in the two be ρ, and let the perpendicular on the tangent plane at P to the confocal (√α 2 + λ, Vb 2 + ∖, Vc s + λ) through P be ω. Then the thickness of the shell at P is μω, and the force at P due to the shell 4πp l uω. Hence the force due to the original shell is dV , abc . .

dv - 7rpμω √(α !> + λ )(^ + λ χ c s + λ) · · , (a) ’

dv being an element of the normal at P. Now if x,y,z be the coordinates of P, we have, by differentiation of

≈ a ∣ y i ∣ ≈ a i

α 2 + λ 0 2 + λc 2 + λ ’

2xdx 2ydy 2zdz ( x i ι y i ι z a ) a s +λi 2 + λc ’ +λ~ ⅞ (a 2 + λ) 2 + (δ 2 + λ) s + (^+λj^ 2 J λ-

Suppose we take dx, dy, dz in the direction of the normal, then dz = dv a ∙i^ κ > & c · » an 4 the last equation reduces to dλ = 2α>dp.

Hence from (a) we get jv 2πpμabcdx

^^√(a 2 + λ)(δ 2 + λ)(c 2 +^λ) -

Integrating this from λ to ∞ , and remembering that the potential vanishes at an infinite distance, we get

V -2τrouxιbc ∕^ , —.... (j8).

-∕ V(α 2 + λ)(5 2 + λ)(c s + λ)

We pass from this to the electrical case by putting for Aκpμabe, which is the mass of the shell, Q, which represents the quantity of electricity on the ellipsoid. We thus get

V=≈- ∕*- — (35) 2 -∕ V(α a + λXδ 2 + λXc 2 + λ) ’ * ’ Ç .

which gives the potential due to a charge Q on an isolated ellipsoid abc at any point on the confocal (x ∕α 2 + λ, √7> 2 + λ, x ∕c 2 + λ). It is obvious that, of the three confocals at P, that is meant which belongs to the same family as (a, b, c), e.g., if (a,δ,c) be an ellipsoid, as opposed to a hyperboloid of one or two sheets, then (s ∕α 2 + λ, *Jb 2 + λ , J c 2 + A) must be an ellipsoid.

If we put λ = 0, we get the value of the potential V∩ at Q · °

the surface. Now — is what we have defined above as v o the capacity of the ellipsoid; we get therefore in the reciprocal of the integral

1 ∕~ α0 dκ

Vo √(α 2 + λ)(δ 2 + λ)(c 2 + λ) ’

an expression for the capacity of an isolated ellipsoid.

In the particular case of an ellipsoid of revolution, the above integral, which is in general an elliptic integral, can be found in finite terms. In the case of a planetary ellipsoid, a = b>c∖ and we find for the capacity

(37),

⅜ιr -«

where e is the least angle whose tangent is - r -f-∙ .

' √α 2 - c 2

If we make c = 0, then e = 0; and the planetary ellipsoid reduces to a circular disc, the capacity for which is there-

^ιCf∕ '

fore —, that is, that of a sphere of the same radius

7r' ’ 1∙571 r (for the capacity of a sphere is obviously equal to its radius). Cavendish had arrived by experiment at the value φT (see Thomson’s Reprint, p. 180), a very remarkable result for his time. It is very easy, by taking the limit of the right hand side of (34), to find the expression for the density at a distance r from the centre of the disc; it is

σ-, — X= · (38).

4πα λ ∕ α s. r a 2τ i √ α 2 - r 2

In the case of an ovary ellipsoid, α = δ<c^ and the • capacity is

2 fc i -a i log f c -+∆2≥∖ .... (39);

-√c 2 -α√

from which several limiting cases may be deduced.

Formula (34), applied to a very elongated ovary ellipsoid, shows us that the density at the pointed ends is very great compared with that at the equator. The ratio of the densities in fact increases indefinitely with the ratio of the longest to the shortest dimension. We have in such an infinitely elongated ellipsoid an excellent type of a pointed conductor.

The effect of a point or an edge on a conductor may be very easily shown by drawing a series of level surfaces, the first of which is the surface of the conductor itself, which has, say, an edge on it. The consecutive surfaces have sharpness of curvature corresponding to the edge, which gets less and less as we recede from the conductor. The level surfaces at an infinite distance are spheres. Tracing, then, any tube of force from an infinite distance, where the sections of all are equal, inwards towards the discontinuity, we see that the section becomes narrower as the curvature of the level surfaces sharpens, and at a mathematical edge the section is infinitely small, and therefore the force is infinitely great. At a mathematical point this is doubly true. At such places the force tending to drive the electricity into the insulating medium becomes infinite. In practice the medium gives way, and disruptive discharge of some kind occurs.

We can find the distribution on a spherical conductor influenced by given forces, such for instance as would arise from rigidly electrified bodies in the neighbourhood.

The method of procedure would be as follows :—Let U be the potential of the rigidly electrified system alone at any point of the sphere. 'Then the problem is to determine a function V, which shall satisfy the equation v 2 V at every point of space, and have the value C — Ü at the surface of the sphere, where C is a constant to be determined by the conditions of the problem. Expand C - U in series of surface harmonics, and let the result be

C-U=γ 0 + γ 1 + γ s + . . . &c. . . . (α ).

Then the value of V is

”Το + Τϊä + ,y2 ä|+ · · . inside the sphere . . (3), and t ' = γ 0 - + 7ι-∣ + 7,-l + .... outside . . (y).

τ r∣ rl

For these evidently satisfy Laplace’s equation, have the given value (a) at the surface of the sphere, and are finite and continuous everywhere. From (0) and (γ), by means of the surface characteristic equation, we can deduce an expression for the density at any point of the sphere, and for the whole charge. If the latter is given we have a condition to determine C; if, on the other hand, the value of the potential of the sphere were given, then this would be the value of C.

The case of two mutually influencing spheres was treated by Poisson in the famous memoir which really began the mathematical theory of electricity. We regret that we cannot afford space for more than a mere sketch of his methods.

Consider the potentials due to the distributions on each sphere. Let a and b be the radii of the two spheres, r and r ’ the distances of any point P from their respective centres, and μ and μ’ the cosines of the angles r and / make with the line joining the centres of the spheres. Since the distributions are evidently symmetrical about the central line, we can obviously expand the potentials due to each distribution in zonal harmonics relative to the corresponding sphere. Hence, if 4πaψ (μ, denote potential due to sphere a at any point inside it, we have

4τrα<∕> ^μ,^=A 0 + A 1 Q 1 + A,Q 2 L∣ 2 + . . , (β ).

The potential at any external point is

A 0 φ+A 1 Q 1 ^ +A s Q l φ∣-t ..... (3), which may be written 4πT <ρ(μ,-'^ .

Similarly we have foi the other sphere

4πiΦ^M. = B 0 + B 1 Q 1 ' L + B s Q' 2 ⅛ + . . . . (γ)

for the potential at any internal, and 4τ- i φiμ l ,- ) for the r ’ k r√

potential at any external point.

The whole potential, then, will be given by v " 4ιr y ≠ (μ, y ) + Φ (∕√, at any point external to both spheres.

Also 'V = 4πa<ρ(μ, + 4π^ Φ (∕>^ inside a; and

V = 4%T- $ (μ,T^ +4τbφ(μ ’ l H^ inside b.

Now, the conditions of the problem require that the values of V in the two last cases shall be constant. Our functions are, there, fore, to be determined by the equations

√'√) + H'4)-' k (

"≠(m,-2-)+⅛(√,≤)- s j which are to be satisfied with obvious restrictions on r and r , in each case. lieverting, however, to the expressions (a), (3), (- y ), &c., we see that we need not solve the problem in the general form thus suggested; for it will be sufficient if we determine the constants A o , A l , &c., B o , B 1 , &c. Now, if we makeμ = l, /-1,—that is, consider only points on the central line,—then Q 1 = l, Q a = l, fee., Qi = L Q2 “L &c∙ i A o , A 1 &c. B o , B 1 , &c., are the coefficients °f —∣ » —J > ^∙c., aj M p i > -∆∣ , &c., in the expressions for the potentials inside the spheres a and b. Hence, if f Ç a ∏d F (ί) den ° te the value9 ° f φ (μ, d) , * , when μ ’ 1 and √ = 1, we need only solve the equations

^⅛) + e⅛ r (Λ) - 4 ) ..

i ⅛√⅛)÷-(0- Γ w '

where we have replaced r and r ’ by their values c-r f and c - r, e being the distance between the centres of a and b. Poisson then eliminates the function F, by choosing a new variable ξ, such that ι ’ ≡ —- » and remarks that we may give to ξ any value between c ~ i

+ a and - a, and therefore we may write r for ξ; we thus have the same variable in both the equations, and F (~— ) which occurs in both may be eliminated. The result is

⅜ i ⅛-√(jS⅛>,∙⅛ · · · ω∙

This is the functional equation on which depends the solution of the problem of two mutually influencing spheres.

Poisson treats very fully the case of two spheres in contact; for which case, taking α = l, the above equation becomes f(r ∖ _ b f( l + ά-r k ⅛ <7& ∕ ∖

⅛ + (l + ft)(l-r)∙' ∖δ+(l + ⅛χi-r)∕ l+⅛-r * , W He finds a solution,

/«-(i+sxiTfj-y —rr 1 —"· ·<»)

0

It is then easy to find F(r), and write down the general expressions for the potential. Poisson goes on to show that the density at the point of contact of the spheres is zero. He finds, for the mean density on the two spheres 1 and b respectively,

A-“..fφμ dt , 1+b J 1 ^*

0

this being, in fact, the value of f (0), a∏dB=-A— Γ

δ(l + δ)∕ 1-1

0

He shows that the calculation of the ratio β of A to B may be reduced to the calculation of the first of these integrals only. For the difference 4∙∏∙δ 2 B - 4πA between the charges on 1 and b he finds the elegant expression

4π 2 δA ιr — cos , l + δ l + δ

from which it follows that the whole charge is always greater on the sphere of greater radius. He then calculates the value of β for various values of b, and its limit for δ = 0, and next the ratio of the densities at the two points diametrically opposite the point of contact, and finds for the mean density on each of two equal spheres in contact A = Λ log 2. He also calculates for this last case the ratio of the greatest to the mean density. In the case of two unequal spheres, the ratio of the greatest density on the smaller to the mean density on the larger.is found for various values of b. He then passes on to investigate the densities for various values of μ.

All these results are compared with the measurements of Coulomb, and found in satisfactory accordance with them. In his first memoir, Poisson considers the case where the distance between the spheres is great compared with the radii; and in a subsequent memoir he considers the case of two spheres at any distance.

Plana (Sur la distribution de V électricité à la surface des deux Spheres, Turin, 1845) extended the calculations of Poisson, using much the same methods. He also calculated approximately the mean densities in the case of several spheres in contact, and arrived at results which agreed satisfactorily with the experiments of Coulomb. For a table of his results, see the end of the first volume of Riess’s Iieibungselectricitat. An account of the work of Roche, who also followed in the footsteps of Poisson, will be ft>und in Mascart, t. i. p. 290 sqq.

The researches of Green led him to a very valuable - synthetical method, by means of which we can construct an infinite number of cases where we can find the electrical distribution. Suppose that we take any distribution whatever of electricity, for which we know the potential at any point, and consequently the level surfaces. Take any level surface, or parts of level surfaces, inclosing the whole of the electricity, and suppose these level surfaces to become actual conducting sheets of metal. Suppose the electrical distribution inside to be rigid, and connect the sheets of metal with the earth, so as to reduce them to potential zero. The sheets will become charged in such a way that the whole potential at every point in them and external to them is zero. Let now U be the potential at any external point due to inside distribution, and V that due to the charge on the sheets, then we have everywhere on or outside the sheets, U + V = 0, or V = - U. Now U is constant at every point of each sheet; hence V is so also. Hence the distribution to which V is due is an equilibrium distribution per se. Removing now our internal distribution, and changing the sign of that on the sheets, we have a distribution of electricity in equilibrium on a set of conductors of known form, the potential of which at any external point is V = U, where U is known. Also the potential V is clearly constant inside every conductor. Hence, applying the characteristic surface equation, we get for the density at any point of any of our conductors the expression σ= dU 4π Clι>

We might make this a little more general, aud state our result thus :—If we distribute on a level surface or surfaces of any electrical system, completely inclosing that system, electricity with surface density at every point

> lki s distribution will of itself be in equil i-brium, and tile potential at any external point ιvill be AeU.

We have given a physical demonstration of this important theorem. The mathematical reader will easily see the application to this case of the general reasoning about the solution of V 2 V = 0, of which we have already given examples. For a simple but interesting case of this general theorem, see Thomson and Tait’s Natural Philosophy, vol. i. § 508.

To Sir William Thomson we owe the elegant and powerful methods of “Electric Images” and “Electric Inversion.” By means of these he arrived, by the use of simple geometrical reasoning, at results which before had required the higher analysis. We shall endeavour to illustrate these by t ’ wo simple examples. λVe do not follow the methods of the author (for which, see his papers), but take advantage of what we have already laid down.

Let A be any point outside a sphere (fig. 12) of radius α, aud centre C. Let AC=∕, and take B in CA

iS » * a i such that CBCA = α 2 , or CB = y; then it is easily proved that, if P be any point on the sphere,

BP α AP ∕'

Hence if E be any quantity of electricity, we have JL-Z≡-o.' AP BP

Therefore, if we place a quantity E of electricity at A, and a quantity - y E at B, the sphere will be a level surface of these two, that, namely, for which the potential is zero. Another level surface of the system is evidently an infinitely small sphere surrounding A. Hence it follows, from the theorem of Green which we have just discussed, that a distribution of electricity on the sphere, the R

density of which is given by (where R is the resultant force due to E and -f E at any point of the sphere), together with a quantity E at A, gives a system in equilibrium, the potential due to which at any point outside tho sphere is the same as that of E at A, and _ f E at B.

It appears, therefore, that the action of the electricity induced on the uninsulated sphere by the electrified point A is equivalent at all external points to the action of -yE at B. The electrified point B is called by Sir William Thomson the electrical image of A in the sphere. It is obvious that the whole charge on the sphere is - y"> and we can very easily find the density at any point.

In fact, resolving along CP, which we know to be the direction of resultant force, the forces due to A and B, we get

—E

e -⅛∞ cpa -⅛ ∞ scpb E Λt* + AP t -∕ 5 ∖ /E ∕∕ 2 + AP 2 -α 1 ∖ (∕ 1 -α 1 )E

“aP 2 \ 2λAP ∕ αAP 2 ∖ 2∕AP^ √~ - αAP 3 ’

···'-<&£ .·<->·

We might have any number of external points and find the image of each. We should thus get a system which might be called the image of the external system. The distribution induced in an uninsulated sphere by such an external system could easily be found by adding up the effect of each external element found by means of its image. Similar methods might also be applied to an internal system. The solution cm be generalized without difficulty to the case where either the charge or potential of the sphere is given.

Suppose the charge Q given; snperpose on the distribution found above a uniform distribution of amount Q + ?E. This will pro-

Q E

duce a constant potential ~+ÿ a∏ over the sphere, and therefore will not disturb the equilibrium. We have thus got the required distribution of the given charge Q under the influence of A. The density of any point is given by

< r =-Q + JL- (Λ→*)E .... (41).

4πa i 4ιraf 4π∙αAP

So far the method of images is simply a synthetical method for obtaining distributions on a sphere. But Sir William Thomson has shown us how to convert it into an instrument for transforming any electrical problem into a variety of others.

f P be any point (fig. 13), 0 a fixed point, and I y be taken in OP such that OP.OP' = α 2 , theu P' is called the inverse of P with respect to 0, which is called the origin of inversion, or simply the origin; α is the radius of inversion. We may thus invert any locus of points into another locus of points, which we may call the inverse of the former.

Let P, Q and P', Q' be any two points and their inverses. Let us supnose that there is a charge E at Q, and a charge E ’ at Q', which is the image of E in a sphere with radius a and centre 0; so that a E '= øqE· Let V and V' be the respective potentials of E and E' at P and P'. Then we have obviously

V' a r N-V~τ ’ where 01’ = »', OP ’ ~r'. It is very easy to show that, if ds, dS, dv, σ, p, be elements of length, surface, and volume, and surface and volume derιsities, and the same symbols with dashes the inverses of these, then we have

<Z√ α* rfS' α 4 ∖ c⅛~ r i ~ a i ∙ dS~ r* kc -!

σ' r 3 a 3 p r s a s 1 z , λv and — = -≡∙-∙τ≈ ∙ 1 — =-s —- s >. (42).

σ a i r ,i · p a i r s i '

also E — L TV r a I

a s0 E ~ a ~ r' ’ V“α = r 7 I

By means of these equations it is easy to invert any electrical system. Take, for example, the case of any conductor in electrical equilibrium; then, since its potential is everywhere constant, it inverts into a surface distribution, tne potential at any point of which distant r' from the origin is by (42) -, C, where C is the constant potential of the conductor. The surface density at any point of the system is found from that of the corresponding point on the conductor by the equation

, α s < r -√J fr ·

Again, if we consider the system thus found, it is obvious that, if we place a quantity - tzC of electricity at the origin, this will make the potential at every point of the system zero, and we have a solution of the case of an uninsulated conductor, whose surface is the inverse of that of the given conductor, under the influence of an electrified point.

As an example of the use of this method, let us invert the uniform distribution on a sphere with respect to an origin on its circumference, the radius of inversion being the diameter of the sphere. The sphere inverts into an infinite plane, touching at the other end A of the diameter through the origin. Let C be the ø

potential on the sphere so that o∙ = 2 "^ ’ "’here d is the diameter. Hence the density at any point P on an infinite plane influenced by a quantity - Cd of electricity placed at a point 0 distant d from it is given by

, d 2 C ^^ 2τ∕ 3 ‘

Again, inverting points inside the sphere, for which the potential is constant, we get the potential due to the distribution on the infinite plane, at points on the other side from the inducing point, the result being

V ’-^, τ which is the same as that due to dC at 0. Hence the potential at a point on the same side as 0 is that due to a quantity dθ placed at O', where 0'A = 0A. 0' is in fact the image of 0. If we write Q

for - Cd, then we get

, - Qd ∖

^X U3).

v '=V- )

These results might of course have been deduced as particular cases of a sphere and point.

Many beautiful applications of these methods will be found in the Reprvnt of Sir William Thomson’s papers and in Maxwell’s Electricity and Magnetism. Two of these are of especial importance. Adopting the method of successive influences given by Murphy (Electricity, 1833, p. 93), and conjoining with it the method of images, Sir William Thomson treated the problem of two spheres. For his results, see Reprint, pp. 86—97. At the end of that paper two valuable tables are given—I. “Showing the quantities of electricity on two equal spherical conductors of radius r, and the mutual force between them, when charged to potentials u and υ respectively II. “Giving the potentials and force when the charges D and E are given.” The ratio of u to V in the first case and of D to E in the second is also given, for which at a given distance there is neither attraction nor repulsion. An interesting experiment on this curious phenomenon is described in Riess, Bd. i. § 186. For an application of dipolar co-ordinates to the problem of two spheres, see Maxwell.

Thomson also applied his methods to determine the distribution on spherical bowls of different apertures. See Reprint, p. 178 sqq. His numerical results on p. 186 are extremely interesting, as affording a picture of the effect of gradually closing a conductor, and are of great value in giving the experimenter an idea as to what aperture he may allow himself in a vessel which he desires should be for practical purposes electrically closed.

It would lead us too far to discuss here the analytical method of conjugate functions, and the allied geometrical method of inversion in two dimensions. A full account of these, with important applications, will be found in Maxwell, vol. i. § 182 sqq.

We shall conclude our applications with a brief notice of a few of the ordinary electrostatical instruments, referring the reader for an account of some others to the article Electrometer.

If two plates be placed parallel to each other, and one of them raised to potential V, while the other is connected with the earth, then there will be certain charges E and F on the two plates. If p and r be the coefficients of selfinduction for A and B, and q the coefficient of mutual induction, then in the present case E -pV, F = g V, and the energy of the distribution is obviously

Q = ⅜EV⅛V 2 , so that the work done by completely discharging the condenser αV 2 . If we suppose the plates very large compared with the distance between them, then we may treat the case, for all points not very near the edge, as if the plates were infinite.

In this case the lines of force are straight, and the number of lines of force which leave any area on A is equal to that of those which enter the opposite area on B. Hence the surface densities on the plates are equal and opposite in sign. Also we clearly have

'-s- i ⅛ («>

For the number of lines of force which cross any unit of area parallel to the plates is constant, and therefore the resultant force is constant at every point between the plates.

It appears, therefore, from (44) that if we make the distance between our plates very small, the density on the inner surface will be very great, and the whole charge on A very great. An apparatus of this kind for collecting large quantities of electricity at a moderate potential is called an accumulator or condenser. One of the first instruments of this kind was Franklin’s pane, which consisted of two sheets of tinfoil pasted opposite each other on the two sides of a pane of glass. There is of course a practical limit to the increase of capacity in such arrangements, because a spark will pass when the insulating medium is too thin. The greater dielectric strength of glass makes it more convenient than air for an insulating medium, and we shall see by-and-by that it has other advantages as well. When the plate A is of finite size there will in general be a distribution of electricity on the back comparable with the charge which A would hold at potential V if B were absent. When the distance between the plate3 is small, by far the greater portion of the capacity is due to the presence of B. Advantage of this principle has been taken in the condensing electroscope of Volta, which is an ordin ary gold-leaf apparatus, except that the knob is replaced by a circular disc on which is placed another disc fitted with an insulating handle; the discs are covered with a thin coat of varnish which serves as an insulating medium. If we connect with either disc, say the lower, a source of electricity of feeble potential V, and connect the upper disc at the same time with the earth, then a large quantity of electricity at potential V collects on the lower disc. Now remove all connections, and lift away the upper disc. The capacity of the lower disc is thereby enormously diminished. Therefore, since the charge is unaltered, its potential must rise correspondingly; and the gold leaves may diverge very vigorously, although a simple connection with the lower disc alone would scarcely have moved them. This instrument is of great use in all cases where we have an unlimited supply of electricity at feeble potential. Sir William Thomson has devised an accumulator of measurable capacity, called the Guard Ring Accumulator, which is a modification of the arrangement we are discussing.

AB (fig. 14) is a flat cylindrical metal box, the upper end of which is truly plane, and has a circular aperture, into which fits, without touching, a plane disc C, which is supported on the bottom of the box by insulating supports, so that its upper surface is in the same plane with the lid of the box. DE is a metal disc which can be moved by a screw through measured distances, always remaining parallel to AB. When desired, C can be pπt in communication with AB. It may then be regarded as forming part of an infinite plate, so that if AB be at potential V, and DE at potential zero, V

then the surface density on C will be equal to-∣^ , where d is the distance between the plates; and if A be the area of C the whole AV

amount of electricity on C is . If now we break the connection between C and the box and discharge the box, we are left with AV

a known quantity of electricity on C, viz. ·

The most usual and for many purposes the most convenient form of accumulator is the Leyden jar. This is merely a glass jar (fig. 15) coated to a certain height outside and inside with tinfoil. The mouth of the jar is stopped with a cork or wooden disc, which serves the double purpose of keeping dirt and moisture from the uncovered glass inside, and of carrying a wire in metallic connection with the inside coating, which passes up through the stopper and ends in a metal knob. If the glass of the jar be very thin, we may find the distribution on the two coatings by neglecting the curvature; the electric density on the inner surface of the two coatings will then be the same as in the case of parallel plates. If, therefore, the inner coating be at potential V, and the outer at potential zero, the density on the inner coating will be —> and that on the outer —‰ . In the particular case we are consider- 4τrti ing the inner coating forms very nearly a closed conductor, so that there will be very little electricity on its inner surface, and there will also be very little on the wire and knob compared with the amount on the surface of the inner coating which is next the glass. We may therefore put SV

for the whole electricity on the inner coating -^∙, where S is the extent of its surface. The capacity C of the jar is then given by

C-A (45).

4πd

Green calculated to a first approximation the effect of the curvature on the capacity, and found that, if It and R ’ be the greatest and least radii of curvature of the inner coating at any point, then the densities on the inner and outer coatings are given by ⅛M⅛÷⅛)} ····<*> and consequently the capacity of the inner coating by ^f+√∕(⅛÷⅛H ∙∙≡

In any case, C being a constant, we have charge E = CV and energy Q = ⅜CV 2 . Hence if we connect the inner coatings of n similar jars, and charge them to potential V, all the outer coatings being at the same time connected with the earth, we have, E and Q representing the whole charge and energy, E -nCV )

Q=-∣CV 2 j (48) ·

If we discharge such a battery of η jars into another of n similar jars, by connecting the knobs together, and the outer coatings to earth in each case, we have, U being the common potential after discharge, nCV = nCU + n'CU

≈ uιd u = n⅛ v * * « ∙∙∙ ' W

There is therefore a loss of energy represented by

JnCV β -⅜(n + n')CU 2 , thatiβ 2(⅛) cvi < δ0 *∙

η

In other words, an ^,th part of the potential energy is lost. When a battery of jars is discharged through a circuit in which there is a fine wire of large resistance, the greater part of the potential energy lost in the discharge appears as heat in the fine wire. Riess made elaborate experiments on the heating of wires by the discharge in this way, and the results of his experiments are in agreement with the fùrmulae which we have just given. (See Heating Effects.)

We may also arrange a battery of jars by first charging • each separately to potential V in the usual way, and then connecting them in series, so that the outer coating of each jar is in metallic connection with the inner coating of the next. In such an arrangement of jars, it is obvious that in passing from the outer coating of the last at potential zero to the inner coating of the first, the potential will rise to nV. When we come to discharge such a series, the electromotive force to begin with is nV, so that for any purpose in which great initial electromotive force is required this combination has great advantages over n jars abreast. The “striking distance,” for instance, i.e., the greatest distance at which the discharge by spark will just take place through air, is much greater. On the other hand, the quantity of electricity which passes is less, being only CV instead of ttCV; the whole loss of potential energy in a complete discharge is, however, the same.

The case which we have been discussing must be carefully distinguished from that of a series of jars charged by “cascade,” where η uncharged jars are connected up in succession as in last case, and the first charged by connection with the electric machine to potential V, while the outer coating of the last of the series is connected to earth, and the rest of the jars insulated. The whole electromotive force in this case is clearly only V, and, if all the jars be similar, the potential difference between the coatings V

in each is —; the charge on the inner coating of the first is C√ V C , *∖Γ2

therefore —’ and the whole potential energy only ⅛ ——. The arrangement is, therefore, not so good as a single jar fully charged by the same machine. It was fancied by Franklin, who invented this method of charging, that some advantage was gained by it in the time of charging, the notion being that the overflow was caught by the successive jars and that electricity was thereby saved. Charging by cascade was treated by Green. Some of the experiments of Riess bear on the matter (vide Mascart, §§ 190, 191), which, after all, is simple enough.

ι In the theory of accumulators, or condensers as they are often called, much stress has been laid on the difference between “free” and “bound” electricity. Toillustrate the meaning of these terms, let us tr.ke a case where the calculations can be carried out in detail.

Suppose we have two concentric spherical shells, an inner, A, and an outer, B. Let the outer radius of A be a, and the inner and outer radii of B be b and c, so that the thickness of the latter is c - b. We shall suppose that we can, when we please, connect the inside sphere with the earth. It is clear that there can never be any electricity on the inner surface of A. Let the charges on the other surfaces in order be E, F, G. Let us suppose in the first instance that A is at potential V, and B at zero. Then we have to find E, F, G. Draw a surface in the substance of B; no lines of force cross it, therefore the whole amount of electricity within is zero. Hence F = - E. Also, considering the external space, which is inclosed between two surfaces of zero potential, we see tha i G = 0. Thus . . . KP ’ since A is at potential V, we have - _ V.

a b E - 5⅛ v - * v (' he " i , -⅛~<,) · (51)·

In this case, then, there is no electrification on the outside of B, <md an electric pendulum suspended there would give no indication.

Let us now connect A with the earth, so that its potential becomes zero; we have now to find the charges and potentials, our datum being that the whole charge on B is - E.

As before, we have F ’ = - E', but G is no longer zero. We have, however, F' + G' — - E. Hence G ’- E' = - E.

E ’ E' G'

Also, since A is at zero potential, _ i - a = o a b c ’

therefore G' =; - F= E ’ =- J*®; G'- —- .

p p+c p+c

The potential of B is — , or c p + c

In this process, therefore, a quantity E - E ’, or ~~ V, of electri-p + c ’

city has flowed away to earth from A, and a quantity N has passed from the inner to the outer surface of B, while the potential has altered, on A from V to 0, and on B from 0 to —~ V p + c

Suppose now we connect B with the earth, thus reducing it to zero potential. Since the charge on A remains the same, and that on the inner coating of B is equal and opposite to it, it follows that now the charges on A, &c., are ~ V, V, 0, where g denotes

; and the potentials of A and B are - V and 0. After another p ^r c c pair of such operations the charges will be &c., and the

~I2 ~I2

potential,-^ V; after a third, charges, Hl? V, &c., and potential, c I c cl

- V. Hence the charges and potentials go on decreasing in geometrical progression. Amounts of electricity flow away from A equal to q∖ τ , q^ y j qy, &c., in the successive operations, and equal amounts of opposite signs are discharged from B. The sum of all these discharges is the whole original charge on A, lor ^ v (1 + f + cl +fcc ∙ , ad ∙ inf^ =-~ V =pV ·

c

Hence by an infinite number of alternate connections we shall finally discharge the jar completely. The electricity which flows out at each contact is called the “free electricity,” and that which remains behind the “bound electricity.” The quantity which we have denoted by ρ is clearly the capacity of a spherical Leyden jar; it increases indefinitely as the distance between the conducting surfaces decreases, and is very nearly proportional to the surface of the inside coating, when the distance is small compared with the radius of either surface.

It is very easy to extend our reasoning to any condenser.

If, in fact, <7 11 , <7 12 , q. li be the coefficients of self and mutual induction for the armatures, then this potential after operating η times as above is ∖-~^~ ) V, the charges, ~÷~) V and . q . VlllW ∖211S22∕

9 ιs (~— ) V and the amounts of electricity which leave 1 ∖5 ’ 11 s ’ 22∕

and 2 in the nth operation are ±<7<7 11 ( ^ 8a -£11~ ) y respectively.

We must not omit one more interesting case. If we have two infinite coaxial cylinders of radii a and δ (b > a), then obviously the potential is symmetrical about the common axis, and Laplace’s equation becomes

1 dV_· dr 2 + r dr - θ'

The integral of this is V = C log r+D. Let the inner cylin be at potential V 1 , the outer at potential V 2 , then

V- (V 1 - V,) - 1 °g. r - + W-Γ Yl⅛ . (62). log α - log 0 log α - log b ' λ

Hence the surface density on the inner cylinder is given by

_ _ 2 dV _ V 1 -V a .

4τr dr , b ’

4πa log - b a and the capacity per unit of length of same is

- 2 ~s · · · < 58 >∙

2α log ä

This result has important applications in the theory of telegraph cables, and to a form of graduated accumulator, invented by Sir William Thomson, and used by Messrs Gibson and Barclay in their experiments on the specific inductive capacity of paraffin (see Maxwell, vol. i. § 127).

ON THE INSULATING MEDIUM.

It has been assumed hitherto that the medium interposed between the conductors in the electric field is in all cases air—the most prevalent of all dielectric media; or, where any other medium actually occurred, as in the case of the Leyden jar, it has been assumed that the result is the same as if the glass were replaced by air. Experimenters soon recognized, however, that the capacity of a Leyden jar depends very much on the quality of the glass of which it is made. But the nature of this action was very little understood, until Faraday showed by a number of striking experiments that the dielectric has a specific function in all phenomena of induction.

Faraday used in his experiments two identical pieces of apparatus, which were virtually two spherical Leyden jars. The outer coating EF (fig. 16) was divided into two hemispheres, which could be fitted together air-tight. The lower hemisphere F was fitted to a perforated stem, provided with a stop-cock G, so that it could be screwed to an air-pump while the apparatus was being exhausted, and afterwards screwed into a foot H. The upper hemisphere was pierced by a tube, into which was cemented a shellac plug B. C is a metal wire passing down through B, which supports the hollow metal sphere D, forming the inside armature, and carries the metal ball A, by means of which D can be charged and discharged. To give an idea of the size of the apparatus, it may be mentioned that the diameters of the inner and outer spheres were 2 33 in. and 3*57 in. respectively. Two jars were made on the above pattern, as nearly alike as possible. The equality of their capacities was tested as follows. Both were filled with air at the same temperature and pressure. Apparatus I. was then charged, by bringing A in communication with the knob of a Leyden jar, while the coating EF was connected to earth. I. and II. were then placed at a moderate distance from each other, as symmetrically as possible with respect to the observer and other external objects, the outer armatures in both cases being in conducting communication, with the earth. The ball of I. was touched by a small proof sphere, the repulsion of which on the movable ball of a Coulomb balance was measured; after a short interval this measurement was repeated. The balls of I. and II. were then brought into communication, and the charge divided between the internal armatures. The ball of II. was immediately tested as before, and then the ball of I. again. Finally I. and IL were discharged and tested for permanent “stern effect.” The result of one such series of measurements was

I. I . . . . 254,250 I; . . . 124 I 1 . . . IL I 0 I 122 . . . . I . . . 2.

Neglecting the slight dissipation of the charge, and taking account only of the “stem effect” in I., we see that the charges on I. and II. after division are represented by 122 and 124, each of which is not far from the half of the whole disposable charge in I., viz., 124*5; so that the capacities of the two jars must be equal. This will perhaps be clearer if we consider what would happen were the capacities unequal. Let the capacities be C and C', the potential of I. before division V, and the common potential after U, the charge on I. Q, and on I. and II. q and q ’ after division. Then Q = CV, y = CU, ∕=C ’ U, and j + ∕=Q,. The indication of the torsion balance is proportional to the charge of the proof sphere, that is (owing to the symmetry of the arrangements), to the potential of the knob with which it was in contact; or at all events this is true if we consider only readings taken from the knob of the same jar, and that is all we shall ultimately want. But (C + C z ) U=≈CV; hence

C ’ V-U C^ U

Hence the ratio of the capacities is equal to the ratio of the excess of the first over the last reading to the last reading, both being taken from the knob of I. Thus, taking the uncorrected values in the above experiment, the ratio of the capacities would be (250 - 124)÷122, t.e. l - 02. By various experiments of this ldnd, Faraday convinced himself of the equality of his two jars. To test the sensibility of his method, he reduced the distance between the lower hemispheres and the ball in II. from - 62 in. to ’ 435 in., by introducing a metal lining. The capacity of II. was then found to be 1'09 (the mean of two observations). He next compared the capacities of the jars when the lower half of the space between the armatures of one of them was filled with shellac. 'Γhe ratio of the capacities was found to be 1 - 5 (mean of several experiments), the shellac jar having the greater capacity.

It appears, therefore, that, other things being equal, the capacity of an accumulator is greater when the insulating medium, or, as it is called, the “dielectric,” is shellac, than when it is air. The ratio of the capacity in the former case to that in the latter is called the Specific Inductive Capacity of shellac. This we shall in general denote by K. According to this definition, air is taken as the standard, and its specific inductive capacity is unity. Properly speaking, we ought to state the temperature and pressure of the air; we may assume 0° C. as our temperature, and the average atmospheric pressure (760 mm.) as our standard barometric pressure.

It is easy to obtain an approximate value of K from the above result for the shellac apparatus. Remembering that the shellac occupies only one hemisphere, and assuming that the lines of force are not disturbed at the junction of the air and shellac, we have, if p denote the ratio of the capacities,

⅛f≈f ’ ∙ andK≈2p-l.

This gives for shellac K = 2 0, the real value being probably greater. Similar experiments gave for glass and sulphur K= 1∙76 and 2∙24 respectively.

Thus the specific inductive capacities of shellac, glass, and sulphur are considerably larger than that of air. Faraday was unable to find any difference in this respect between the different gases, or in the same gas at different temperatures and pressures, although he made careful experiments in search of such differences.

It would lead us too far to discuss in detail the precautions taken by Faraday to remove uncertainty from his experimental demonstration of the existence of a specific dielectric action. The reader will find a minute description in Faraday’s own surpassingly lucid manner in the eleventh series of the Experimental Researches.

His discovery of the action of the medium led Faraday to invent his well-known theory of the dielectric. According to him, the fundamental process in all electrical action is a polarization of the ultimate particles of matter; this polarization consists in the separation of the positive and negative electricities within the molecules, exactly as the two magnetic fluids are supposed to separate in the theory of magnetic induction. In this view a dielectric is supposed to consist of a number of perfectly conducting particles, immersed in a medium or menstruum, which is either a non-conductor or a very imperfect conductor. When electrical action starts, the two electricities separate in the molecules; but, in the first instance at least, there is no interchange of electricity between different molecules.

Faraday assumed that the electrical action is propagated from molecule to molecule by actions whose sphere of immediate activity is very small. He denied the existence of “action at a distance,” and regarded his results about induction in curved lines as at variance with it. Thomson showed, however, that Faraday’s results were perfectly consistent with the theory of action at a distance, provided the polarization of the dielectric be taken into account, and that the mathematical treatment of the subject is identical with Poisson’s theory of induced magnetism. The theory of action at a distance as applied to this subject will be found under Magnetism. Helmholtz, whose memoirs we ha - already mentioned, takes this view of the matter. We do not propose to follow Faraday’s theory any further at present; its main features are involved in Maxwell’s theory, to which we shall afterwards allude.

W. Siemens examined and confirmed the conclusions of Faraday. He used voltaic electricity in comparing the capacities of condensers. By means of a kind of self-acting commutator (Sclbstthiitige Wippe), the armatures of the condenser were connected alternately with a battery of Daniell’s cells and with each other; so that the condenser was charged and discharged about 60 times per second.

Figure 17 gives a scheme of the arrangement. F and G are two insulated metal screws, with which the vibrating tongue E of the W ιppe comes alternately into contact; CD and AB are the armatures of the condenser, H the battery, and K the galvanometer. Theory indicates, and experiment confirms, that the deflection will be the same whether the galvanometer is put in the charge or in the discharge circuit. The former arrangement is that indicated in the figure.

The amount of electricity which flows through the galvanometer each time the condenser is charged, is proportional to the product of the capacity C of the condenser and the electromotive force E of the battery. E is proportional to the number of cells in the battery. If, therefore, the speed of the Wippe be constant, the galvanometer deflection, or its sine or tangent as the case may be, will be proportional to EC. By varying E and C independently, we can verify the laws that regulate the charge of condensers. If we keep E the same, and the speed the same, we can compare the capacities of two condensers, or of the same condenser with two different dielectrics, and thus find the specific inductive capacities of various substances with respect to air. Siemens found that C is independent of E, and concluded that the effect of solid dielectrics on the capacity of a condenser is not to be explained by a penetration of the electricity into the dielectrics. We shall give some of his values of the specific inductive capacity farther on.

Gaugain studied the effect of the insulator on the capacity of condensers. He used in his researches the discharging electroscope (see art. Electrometer), an instrument which does not at first sight look likely to lead to very accurate results, but which seems to have worked satisfactorily in his hands. Many of Gaugain’s results concerning the gradual increase of the charge are very interesting; their bearing on theory is difficult to estimate, however, owing to the mixture of effects due to surface and body conduction. His results concerning the “limit ing” value of the specific inductive capacity are at variance with those of subsequent experimenters who have worked with more delicate instruments.

In their experiments on the specific inductive capacity of paraffin, Gibson and Barclay employed a method due to Sir William Thomson, in which an instrument called the Platymeter is used in conjunction with the quadrant electrometer. They found for the specific inductive capacity of paraffin 1 - 97, and showed that this value alters very little, if at all, with the temperature.

The most extensive measurements of this kind that have been made of late are those of Boltzmann and Schiller. Boltzmann used a sliding condenser, whose plates could be placed at measured distances apart. Plates of different insulating materials were introduced between the parallel plates of the condejιser, so as to be parallel with them and at different distances from one of them.

According to the mathematical theory, the capacity of the condenser is independent of the position of the plate, and varies inversely as m - n + g , where m is the distance between the plates of the condenser, and n the thickness of the plate of insulating material whose specific inductive capacity is K. In other words, the plate may be supposed replaced by a plate of air of thickness g-. If therefore λ denote in absolute measure the reciprocal of the capacity of the condenser, then

λ = G^wι-n + ~) , ∙

where G is a constant. The capacity of the condenser was measured by charging it with a battery of 6 to 18 Daniell’s cells, and then dividing its charge with the electrometer. One pole of the battery and one armature of the condenser are connected to earth. The other pole of the battery is first connected with the electrode A of the electrometer, whose other electrode B is connected to earth. Let the reading thus obtained be E, then E is proportional to the potential of the battery pole. The condenser is next charged by connecting its insulated armature with the battery; the battery connection is then removed, and the electrode A of the electrometer, which has meanwhile been connected with the earth, is now connected with the condenser. If C be the capacity of the condenser, C' that of the electrometer (in certain cases artificially increased), we have, if F be the common potential of the condenser and connected parts of the electrometer, (C + C')F = CE, and n _ FC* E-F 1

c ~ E-F ’ or λ - F ’σ’

But F is proportional to the second reading of the electrometer, hence λ is known in terms of C'. As only relative measures are wanted, C' is not required. Boltzmann made a variety of experiments, all of which confirmed the theory, and showed the applicability of the above formula.

If we make three measurements, first with the plates at distance m 1 ∙, secondly, at distance m 2 , with only air between in each case; and thirdly, at distance rn 3 , with an insulating plate of thickness n between, we have, if λ x , λ 2 , λ 3 be the corresponding values of λ,

G = —— λl - , and ~ = (λ 3--- λl -wι 3 + m, + n)÷n.

The advantage of this procedure is that only differences of wι 1 , rn 2 , rn 3 come in, and no absolute length has to be measured. Measurements were also made with condensers, in which there was no air between the armatures and the insulating plates; in them the armatures were formed by means of mercury. To give an idea of the agreement of the results by different methods, we give K for paraffin as determined on plates of different thickness; with the ordinary condenser, K = 2 28, 2 - 34, 2 - 31 for plates I., 11., and III.; and K = 2 - 31, 2 - 33 for plates I. and II. used with mercury armatures.

Boltzmann convinced himself that, in the case of ebonite, paraffin, sulphur, and rosin, the time during which the condenser was charged was without sensible influence. He found that the result was the same whether the charge was instantaneous or lasted for a considerable time. The case was different with the imperfect insulators, glass, stearine, and gutta percha, for which he has given no results. To test still farther the influence of the time, Boltzmann measured the attraction between a sulphur and a metal sphere—first, when the latter was charged continuously positive or negative, and, secondly, when it was charged positive for ^⅜ π th θf a second, negative for the next 7 ⅜7 j ∙th, and so on; he found the attraction to be the same in both cases, provided the charges without respect to sign were equal. This experiment establishes beyond a doubt the existence, in the case of sulphur, of a specific dielectric action, which is fully developed in less than 3f ⅜ δ th of a second. From experiments of this kind values of K were deduced, which agreed fairly well with those obtained by other methods. A very important result which he obtained was, that for a certain crystalline sphere of sulphur the values of K were different in the directions of the axes, being 4'773, 3 - 970, and 3 ’ 811 respectively. The result realizes an expectation of Faraday.

Schiller employed two methods—the method of Siemens, which we have already described, in which the duration of charge was from j⅛th to ^ 1 σ th of a second, and the method of electrical oscillations devised by Helmholtz. In the latter method K is given by the equation K = (T 2 - T o 2 ) ÷ (T' 2 - T 0 2 ), where T o , T, T', are the periods of oscillation of a certain coil, firstly, by itself, secondly, when connected with an air-condenser, and thirdly, with the same condenser when the air is replaced by the insulator to be tested (see below, p. 82). In this method the duration of charge varied from ^<∏½∏ i ∙tlι to g0 1 00 th of a secohd.

The following table gives some of the results of Boltzmann and Schiller:— [table]

The first column of Schiller’s results was obtained by Siemens s method, the second by the method of oscillations. It will be seen that the shortness of the time of charge has affected the value of K in the last column, reducing it considerably in all cases. Boltzmann’s results are on the whole the largest obtained by any physicist; he attributes this to the care with which he constructed his plates. Gibson and Barclay found 1’97 for paraffin, and Siemens 2 9 for sulphur.

Among the more recent researches on the theory of dielectrics may be mentioned those of Rood, whose results for crystals are interesting, and Wiillner, who has studied the course of induction when the charge is maintained for a considerable time.

There are very few fluids which are sufficiently good insulators to allow an easy determination of their specific inductive capacity. Measurements have, however, been made by Silow. He used (1) Siemens’s method, and (2) a method in which he observed the deflection of a quadrant electrometer corresponding to the same potential, first, when the quadrants were filled with air, and secondly, when they were filled with the fluid to be examined; the ratio of the latter deflection to the former is the specific inductive capacity of the liquid.

The instrument actually used was a glass vessel, inside which were pasted pieces of tinfoil corresponding to the quadrants of

Thomson’s electrometer. The shape of the needle was also slightly different. A fine silver wire replaced the bifilar suspension, and the deflections were read off by means of a scale and telescope. The needle and one pair of quadrants were connected with the earth, and the other pair of quadrants charged to a constant potential by connection with a battery. The results were for oil of turpentine by method (1), 1'468; by (2), 1'473; for a certain specimen of petroleum, by (1), 1'439; for another specimen, by (2), 1'428; for benzol, by (1), 1'483.

In the researches in which Siemens’s method was used, the speed of the commutator was varied considerably, but no effect was thereby produced on the value of K, which is therefore, within certain limits at least, independent of the duration of the charge.

Perhaps the most important of all the recent additions to our knowledge in this department is due to Boltzmann, who has succeeded in detecting and measuring the decrease of the specific inductive capacity of gases when rarefied.

The principle of his method is as follows. Suppose we have an ordinary air-condenser inside a receiver, which we can exhaust at will. Let one of the armatures A of the condenser be connected with a battery of a large number n of cells (Boltzmann used about 300 Daniell’s), while the other armature B is connected with the earth. If we now insulate B, and if the condenser does not leak, then on connecting B with the electrometer no deflection will be indicated. If, however, we increase the number of cells by one, the potential of A will increase from up to (π+ l)p, while that of B will rise from 0 to an amount which is proportional to ρ. Let the corresponding electrometer reading be β. Suppose, now that we altered the specific inductive capacity of the gas from K 1 to K 2 , both armatures being insulated, A originally at potential up, and B at potential zero; the potential of A will, by the mathematical theory, become pnp, while that of B remains zero. If now we reconnect A with the battery of n cells, the potential of A becomes again np. If we then connect B with the electrometer we shall get a deflection a. proportional to np (1 - j ⅞); hence we have —∣ = n^l - .

Let us now assume, what experiment shows to be the case, that the increase of K is very nearly proportional to the pressure, then, b 1 and b i denoting the manometric reading in millimetres corresponding to K 1 and K 2 , we may write κ *= e (,+ ⅛)' κ ∙= c (1 + -J⅛)∙

Here λ is a constant, the meaning of which is very simple, if we assume our law of proportionality to hold up to absolute vacuum; in fact, 1 + λ is in that case the specific inductive capacity of the gas at 760 mm. pressure, at the temperature t of observation, , and l + λ(l + α∕) is the corresponding coefficient at 0° C. The formula written above becomes therefore λ - βw(5 1 -Z> 2 ) ‘

In this way Boltzmann arrived at the following values for Ji∖, at 760 mm. pressure, and temperature 0° C. :—for air, 1Ό00295; carbonic acid, 1Ό00473; hydrogen, 1Ό00132; carbonic oxide, 1Ό00345; nitrous oxide, 1Ό00497; olefiant gas, 1Ό00656; marsh gas, 1Ό00472. These results are of great importance in connection with the electromagnetic theory of light.

Residual Discharge.

When an accumulator, whose dielectric is glass or shellac, is charged up to a moderately high potential, and one armature insulated, a gradual fall of the potential occurs. This fall is tolerably rapid at first, but it gets slower and slower till at last it reaches a certain limit, after which it remains sensibly constant for a considerable time. This fall is not entirely due to loss by conduction or convection of the ordinary kind, for we fitιd that if an accumulator that has been charged to potential V, and has been allowed to stand till the potential has fallen considerably, be again charged up to potential V, then the rate of loss is much less than before, being now very nearly constant, and not far from the limit above mentioned It would appear, therefore, that this constant limit, which on favourable days is very small, represents the loss due to convection and conduction in the usual way, and that the larger varying loss is due to some other cause. When an accumulator, let us say a Leyden jar, has been repeatedly charged up to potential V, until the rate of dissipation has become constant, we shall say that it is saturated. If we discharge a saturated jar, by connecting the knob for a fraction of a second with a good earth communication, and then insulate the knob, the outer coating being supposed throughout in connection with the earth, we find that the instant after the discharge the potential of the knob is zero; after a little, however, it begins to rise, and by and by it reaches a value which is a considerable fraction of V, and has the same sign. This phenomenon justifies the assumption we made as to the peculiar nature of the variable loss of potential experienced by a freshly charged jar. The charge which reappears in this way subsequent to the instantaneous discharge is called the residual discharge. If at any time during the appearance of the residual charge the jar be discharged, the potential of the knob becomes for a short time zero, but begins to rise again; and this may be repeated many times before all trace of charge disappears. Faraday made a variety of experiments on the subject, and established that whenever a charge of positive electricity disappeared or became latent in this way, an equal negative charge disappeared in a similar way. He concluded that the cause of the phenomenon was an actual penetration of the two electricities (Exp. Res., 1245) by conduction into the dielectric. This is not the view which is favoured by the best authorities of the present day; it is indeed (see Maxwell, Elect, and Mag., vol. i. § 325) at variance with the received theories of conduction, and alike untenable, as far as we know, whether we adopt the theories of Weber, of Maxwell, or of Helmholtz. Faraday established that time was a necessary condition for the development of the phenomenon; and he was thus enabled to eliminate its influence in the experiments on the specific inductive capacity of sulphur, glass, and shellac. The phenomenon is most marked in the last of these; and in spermaceti, which relatively to these is a tolerably good conductor, the phenomenon is very marked, and develops very rapidly.

Kohlrausch studied the residual disharge in an ordinary Leyden jar, in a jar whose outside and inside coatings were at one time quicksilver and at another acidulated water, and in a Franklin’s pane, one side of which was coated with tinfoil in the usual way, while the other was silvered like a piece of looking-glass He showed, by taking measurements with an electrometer and a galvanometer, that the ratio of the free or disposable charge to the potential is constant. By the disposable charge is meant the charge which is instantaneously discharged when the knob of the jar is connected with the earth. This ratio is the capacity of the jar, and it appears that it is independent of the ’residual” or “latent charge. He showed that the “latent” charge is not formed by a temporary recession of the electricity to the uncovered glass about the neck and upper part of the jar; and that it does not to any great extent depend on the material used to fasten the armature to the glass, or on the air or other foreign matter between them. On the other hand, his results led him to suspect that the “latent” charge depended on the thickness of the glass, being greater for· thick plates than for thin. This conclusion has been questioned, however. He separated by a graphical method the loss by latent charge from the loss by conduction, &c., and found that the amount of charge which becomes latent, or, which amounts to the same thing, the loss of potential owing to the forming of latent charge in a given time, is proportional to the initial potential so long as we operate with the same jar.

Kohlrausch recognized the insufficiency of Faraday’s explanation of the residual charge, and sought to account for it by extending Faraday’s own theory of the polarization of the dielectric. The residual charge is due according to him to a residual polarization of the molecules of the dielectric, which sets in after the instantaneous polarization is complete, and which requires time for its development. This polarization may consist in a separation of electricity in the molecules of the dielectric, or in a setting towards a common direction of the axes of a number of previously polarized molecules, analogous to that which Weber assumes in his theory of induced magnetism. It is easy to see that such a theory will to a great extent account for the gradual reduction of the potential of a freshly charged jar, and the gradual reappearance of the residual charge.

If the charge, and consequently the potential, of the jar were kept constant at Q o , the residual charge tends to a limit ρQ 0 (p const.), Kohlrausch assumes that the difference rt~pQ< i between the residual charge actually formed and the limit decreases at a rate which is at each instant proportional to this difference, and furthermore, to a function of the time, which he assumes to be a simple power. In any actual case, where the jar is charged and then insulated, the charge varies, owing to conduction, &c., and to the formation of residual charge, so that the limit of r» is continually varying, and we must write Q l for Q o , Q, denoting the charge at time t. The equation for residual charge is then f y z (∏-pQι) - ~ bV(p<b - rt) ·

From tlιis he deduces the formula

∕ -Lt∙+ 1 ∖

r< = p i Q∣ - Q 0 e -→ I > which he finds to represent his results very 7 closely, rn has very nearly the same value ( - 0 - 5744, or -⅜ nearly) in all his experiments, p had the values 0’4289, 0’5794, 0’2562; and b 0 - 0397, 0 - 0223, 0'0446 in his three cases.

Kohlrausch called attention to the close analogy between the residual discharge and the “elastic recovery” (elastische Nachwirkung) of strained bodies, which had been investigated by Weber in the case of a silk fibre, and which has of late excited much attention. The instantaneous strain which follows the application of a stress is analogous to the initial charge of the jar, and the gradually increasing strain which follows to the gradual formation of the latent or residual charge. The sudden return to a position near that of unconstrained equilibrium corresponds to the instantaneous discharge, and the slow creeping back to the original state of equilibrium to the slow appearance of the residual discharge. Another analogy may be found in the temporary and residual or subpermanent magnetism of soft iron or steel. If we wish to make the analogy still more complete, we have only to introduce the permanent polarity of tourmaline, the permanent set of certain solids when strained, and the permanent magnetism of hard steel. The phenomena of polarization furnish yet another analogy.

In justifying the introduction of a power of the time into his equation for the residual discharge, Kohlrausch makes the important remark that the time which a residual charge of given amount takes to reappear fully may be different according to the way that charge is produced. The charge reappears more quickly when it is produced in a short time by an initial charge of high potential, than when produced by a charge of lower potential acting longer. He suggests that the same thing may be true of elastic recovery. He does not allude to the fact (possibly he was unware of it) that two residual charges of different sign may be superposed and reappear separately, although the possibility of this is to a certain extent involved in his remark. The analogous elastic phenomenon has recently been observed by F. Kohlrausch.

Maxwell has shown that phenomena exactly like the residual discharge would be caused by conduction in a heterogeneous dielectric, each constituent of which by itself has not the power of producing any such phenomenon, so that the phenomenon in general might be due to “heterogeneity” simply.

Hopkinson has lately made experiments on the residual discharge of glass jars. He observed the superposition of residual charges of opposite signs, and he suggests theories analogous to those of Kohlrausch and Maxwell. He finds that his results cannot be represented by the sum of two simple exponential functions of the time, and concludes, therefore, that heterogeneity must be an important factor in the cause of the phenomenon.

The polarities of the different silicates of which the glass is composed rise or decay with the time at different rates, so that during insulation the difference of potential between the armatures E would be represented by a series 2” A r e~^ rt ∙ If, therefore, we charge a jar positively for a long time, and then negatively for a shorter time, the second charge will reverse the more rapidly changing polarities, while the sign of the more sluggish will not be changed; when, therefore, the jar is discharged and insulated, the first-mentioned polarities will decay more quickly at first and liberate a negative charge, and, finally, as the more sluggish also die away, a positive charge will be set free. Hopkinson also made the important observation that agitation of the glass by tapping accelerates the return of the residual discharge.

ON THE PASSAGE OF ELECTRICITY THROUGH BODIES.

We have hitherto supposed electricity to be either immovably associated with perfectly non-conducting matter, or collected on the bounding surfaces of conducting and non-conducting media in such a way that the force tending to cause it to move is balanced by an invincible resistance. We have now to consider what happens when there is a finite unbalanced resultant force at any point in a conducting medium. If a conducting sphere of radius a be charged with Q units of positive electricity, its potential will be —. Connect this sphere by a long thin wire, whose capacity may be neglected, with another uncharged sphere of radius b, then we know that the potentials of the two spheres become equal; and since what we call electricity is subject to the law of continuity, the whole charge on the two spheres must be the same as before. Hence if U be the common potential, we must have U = · It ap pears, therefore, that the potential of a has fallen by —Q, and an amount Q of positive electricity has passed from a to δ, and also a ^-^th part of the electric potential energy has disappeared. In accordance with our hypothesis that electricity obeys the law of continuity like an incompressible fluid, we explain this transference of electricity by saying that an electric current has flowed through the wire from the place of higher to the place of lower potential. We define the intensity or strength C of the current as the quantity of electricity which crosses any section of the wire in unit of time.

Owing to the law of continuity the current intensity is of course the same at every point of a linear conductor.

In the case which we have just given, the whole transference takes place in so short a time that we cannot study the phenomenon in detail. It is obvious that C will vary rapidly from a large initial value, when the difference between the potentials of the spheres is , to zero when they are at equal potentials. It is possible, by replacing the wire by wetted string or other bad conductor, to prolong the duration of the phenomenon to any extent, so that C should vary very slowly; and we can imagine cases where C would remain constant for a long time. Machines for producing a continuous or “steady” current have been invented in considerable variety, the first of the kind having been the Pile of Volta. Of such machines we shall have more to say when we come to discuss Electromotive Force. We have seen, in the case of our spheres, that the passage of the electric current was accompanied by a loss of potential energy. The question thus arises, what becomes of the energy after the current dies away, and the equalization of potential is complete I This leads us to look for transformations of energy depending on the electric current, or, in other words, to look for dynamical effects of various kinds due to it. Accordingly we find the passage of the electric current accompanied by magnetic phenomena, sparks, heating of the circuit, chemical decompositions, mechanical effects, Ac. All these are observed in the discharge of the Leyden jar and other electrostatic reservoirs of potential energy. Exactly similar effects, some more, others less powerful, are observed accompanying the current of the voltaic battery and other machines which furnish a steady flow of electricity. In all such cases we have (1) a source of energy, (2) a flux of electricity, (3) an evolution of energy in different parts of the circuit. We reserve the consideration of (1) for the present, as being the most difficult, and devote our attention to (2) and (3).

Ohm’s Law applied to Metallic Conductors.

We have already seen how to measure the strength of an electric current in a linear conductor. According to the definition we gave above, the unit current strength would be that for which a unit of electricity passes each section of the conductor in unit of time. If the unit of electricity is the electrostatic unit, this is called theelectro-statical unit of current. We have supposed above that the current consists in the transfer of a certain amount of + electricity in a certain direction, which we shall call the positive direction of the current, and this for most purposes is convenient. We must remember, however, that no distinction can be drawn between the transference of + Q units of electricity in one direction and the transference of - Q units in the opposite direction; for we have no experimental evidence on which such a distinction can be founded.

We may measure the current by any one of its various effects. The method most commonly used, both for indicating and measuring currents, is to employ the magnetic effect. According to Oersted’s discovery, a magnetic north pole placed in the neighbourhood of a straight current is acted on by a force such that, if the pole were to continually follow the direction of the force, it would describe a circle round the current as an axis, the direction of rotation being that of the rotation of a right handed cork-screw which is traversing a cork in the positive direction of the current. If, therefore, we have currents of different strength in the same wire, the forte exerted on a magnet which always occupies the same position relatively to the wire will be a measure of the current. The force exerted on the magnet may be found by balancing it against known forces, or by allowing the magnet to oscillate under it and finding the time of oscillation. It is easy, by applying the law of continuity to multiple circuits, to verify that the measure of current intensity thus got is proportional to the electrostatic measure.

Thus let AB (fig. 18) be a circuit splitting up into two exactly similar branches BCDG, BEFG, and uniting again at G. Then, since electricity behaves like an incompressible fluid, it is 0bviou3 that any current of intensity C in AB will split up into two currents each of intensity ⅜C in CD and EF. By placing a magnet in similar positions at the same distance with respect to AB, CD, and EF, it will be found that the magnetic action in the last two positions is just half that in the first.

The appropriate unit in magnetic measurements of current intensity is that current which, when flowing in a circular arc of unit radius and unit length, exerts unit of force on a unit north pole placed at the centre of the arc, the unit north pole being such that it repels another equal north pole at unit distance with unit force. This is called the electromagnetic unit of current intensity. Unless thf contrary is stated, all our formulae are stated in terms of this unit.

To facilitate the detection and measurement of currents by magnetic means, an instrument called a galvanometer is used. It consists of a coil of wire, of rectangular, elliptical, or circular section, inside which is suspended a magnetic needle, so as to be in equilibrium parallel to the coil windings under the magnetic action of the earth, or of the earth and other fixed magnets. When a current passes through the coil a great extent of the circuit is in the immediate neighbourhood of the magnet, and the magnetic action is thus greatly accumulated. See article Galvanometer.

If we connect two points A and B of a homogeneous linear conductor, every point of which is at the same temperature, by two wires of the same metal to the electrodes of a quadrant electrometer, then, if a steady current C (measured in electrostatic units) be flowing from A to B, we shall find that the potential at A is higher than that at B by a certain quantity E, which we may call the electromotive force, between A and B, and we may suppose E for the present to be measured in electrostatic units. If we examine the value of the ratio ≈, for different ρosi tions of the points AB, we shall find that it varies directly as the length of linear conductor between A and B, provided the section of the conductor is everywhere the same. If we try wires of different section, but of the same length and the same material, we fiud that is inversely proportional to the sectional area; in fact we may write c~ r -4 z w, where l denotes the length of the wire, ω its section, and k a constant depending on its material, temperature, and physical condition generally. This is Ohm’s law.

In whatever unit measured, R is called the resistance of the conductor. The unit of resistance can always be conceived as established by means of a certain standard wire. The unit of electromotive force is then such that if applied at the end of the standard wire it would generate a unit current in the wire, lhe constant k is called the specific resistance of the material of which the wire is made; it is obviously the resistance of a wire of the material of unit length and unit section.

In the electrostatic system of unitation the unit of E is the work done by a unit particle of + electricity in passing to infinity from the surface of an isolated sphere of radius unity charged with an electrostatic unit of + electricity. The dimension of E is ^QL^ 1 2 ’ where [Q] is the dimension of the electrostatic unit of quantity

(see p. 22), — [^L^M 2 T J. Hence the dimension of E is [l^M’T 1 ]. The unit of C we have already discussed; its dimension is [^QT 1 J “ ^L 5 M 1 T * J From these results, and equation (1), it follows that the dimension of R is ^L^ 1 τJ, i.e., that of the reciprocal of a velocity. We shall show hereafter that, if C be measured in electromagnetic units, its dimension is [l⅛⅛t^ , ] 5 hence that of Q is ^L⅛I⅛J, the unit of Q being the quantity of electricity conveyed across any section by the unit current. Also ECT — work done in time T in conveying C units of + electricity from potential V + E to potential V, whence [ECT] = dimension of energy =[p, s MT~']∙ Hence [e]” [lWt^ 2 ]. In this case then [r] = [l , Γ *] î so that in electromagnetic measure R has the dimension of a velocity.

We can put the equation (1) into another form, which suggests at once the generalization of Ohm’s law for any conductor. Con sider two points P and Q on a linear conductor, at a distance dx from each other, x being measured in the direction of the current. Let the potentials at P and Q be V and V + dV, then E= -dV. If u denote the current per unit of area of the section, then C = iiω kdx and since l=dx we have R——. Substituting these values in (1) we get 1 dγ X u ~~ kdx ~ k &·

where X is the component electric force at P in the direction of the current. Since the electric current is of the nature of a flux, it is determined at any point of a conductor by the flux components uνw, representing the quantities of electricity which in unit of time cross three unit areas perpendicular to three rectangular axes drawn through P. If X,Y,Z be the components of the electric force at P, then the general statement of Ohm’s law for a homogeneous isotropic conductor is y V 7

U -k V -k W -k

In such a conductor the resistance of a small linear portion of given dimensions, cut out of the substance any where or any how, will Ire the same. It is conceivable, however, that the resistance of such a small portion would be different if cut in different directions at any point, in which case the conductor would be aeolotropic. The most general statement of Ohm’s law would then be u=r 1 X + p 3 Y + ⅞Z )

v≈q 3 X + r 3 Y+p 1 Z J (4), w=p i X + q 1 Y + r 3 Z )

where r 1 , &c., p 1 , &c., q l , &c., are constants for any one point. If they are the same for all points, the body is said to be homogeneous; if they vary from point to point, the body is said to be heterogeneous. If we may liken our conductor to an arrangement of linear conductors (see Maxwell, §§ 297, 324, vol. i. ), then it may be shown that the skew system of (4) becomes symmetrical, inasmuch as ρ i ≈q l , p 2 ≈g i , p 3 ≈q 3 ∙ The great majority of the substances with which the electrician has to deal are, however, isotropic; and unless the experiments of Wiedemann on certain crystals point to aeolotropic conduction, we do not know of any case which has been experimentally examined. The reader will find interesting developments of the subject in Maxwell, vol. i. § 297 sqq.

A very important remark to be made with regard to the equations (4) is that, being linear, the principle of superposition applies. Thus, if u,v,w be the current components due to electric forces X,Y, Z and u'∖vl,wl similar components for X ’,Y ’, Z ’, then the current for X + X ’, Y + Y', Z + Z' is given by u + u', v + √, w+w∕. It is obvious, moreover, that (4) are the most general equations that can be written down to connect current with electromotive force, subject to the condition that the currents due to superposed electric forces are to be found by the superposition of the currents due to the separate forces.

Besides the equations (4), u,v,w are subject like any other flux components to an equation of continuity. This equation, investigated in the usual manner, is + (6 ), dx dy dz dt where ρ is the electric volume density at the time t. At a surface of discontinuity (5) must be replaced by

(u-u ’ ')l + (υ-v f )m + {w-w)n-^=Q . . (6), where u,t>,w, and u',v ’,w are components of flux on the first and second sides of the surface, l,m,n the direction cosines of the normal drawn from the first to the second side, and σ the electric surface density at time t.

If ' we consider the particular case of homogeneous isotropic r r t , t1 t γ dN v dV _ dV

media, and suppose further that Λ = - —> Y = - Z-- - »

these equations reduce to

<*W,fV.rfW-Λ ,-x c⅛ 2 df dz i k dt u ' ’

, 1 dV 1 J dN i -dσ .

and k 1 dv + ⅛ 8 dv ~dt (8) ·

In the last equation V 1 and V 2 are the potentials on the two sides of a boundary between media of specific resistance ⅛ 1 and k i .

In the particular case of stead} motion, the right-hand sides of (7) and (8) are zero. The analytical treatment of problems about steady currents is therefore precisely analogous to that of problems about electrostatical equilibrium, steady flow of heat, hydrodynamics, &c.: to every solution in one such physical subject corresponds a solution in each of the others. Many valuable details on this subject are to be found in Thomson’s papers on electrostatics and magnetism.

The consequences of Ohm’s law have been followed 5 out mathematically, and verified in a variety of cases. We shall notice a few which are interesting, either from the accuracy of the experimental results, or from the interest or practical importance of some method or principle involved.

In the case of a steady current in a uniform linear conductor, say a wire, it is obvious that the potential must fall uniformly in the direction in which the current is flowing. Hence, if we suppose the wire stretchèd out straight, and erect at different points lines perpendicular to it, representing the potential at each point, the locus of the extremities of these lines will be a straight line.

This may be arrived at by integrating equation (5), which bed 2 V

comes in tlιis case ^⅜=0, x being measured along the wire sup- 5osed to be straight; we thus get for the potential V, at any point istant x from the origin, at which potential is V o ,

V = V 0 --z (9).

ω

If V be taken as ordinate, this represents a straight line, the Cλ^

tangent of whose inclination to the a;-axis is , or - t<Jc.

ω

We cannot apply Ohm’s law at the junction of two different substances. The condition of continuity of course applies; in other words, if the flow has become steady, the current is the same at all points of the circuit, whether homogeneous or not. We shall see, when we come to discuss electromotive force, that there is a constant difference between the potentials at two points infinitely near each other, but on opposite sides of the boundary between two conductors of different material. If we knew this potential difference for each point of heterogeneous contact in the circuit, we could draw the complete potential curve for the circuit by applying Ohm’s law to each conductor separately. The diagram (fig. 19) represents (on the contact theory, as held by Ohm, see Origin of Electromotive Force) the fall of potentials and the discontinuities in a voltaic circuit, consisting of zinc, water, and copper, in which the current flows from Cu to Zn across the junction of the metals. We assume for the present that Ohm’s law applies to the liquid conductor.

Let us denote by V q , V k , &c. the potentials at Q and R, &c., or what is the same thing, the ordinates BQ, BR, &c., in our diagram. Then applying Ohm’s law to the homogeneous parts of the circuit, we have V v -V ς = CR', V a - V s = CS, V i - V p — CR' ’, where R', S, R", denote the resistances of the zinc, the water, and the copper respectively. Now, denoting V v - V u , the potential difference, or as it is sometimes called, the “contact force” between Zn and Cu by E zc , and so on, let us add the above three equations; we thus get

E ~ E zc + Eaz + Ec a ~ C(R' + R z, + S).

Here E is called the whole electromotive force of the circuit, being the sum of all the discontinuities of potential, taken with their proper signs, or, what is equivalent to the same thing, the whole amount of work which would be done by a unit of + electricity, in passing round the whole circuit once, supposing it to get over the discontinuities without gain or loss of work. Defining E in this way, we may extend Ohm’s law to a heterogeneous circuit, the resistance R being now the sum of all the resistances of the different parts, or the whole resistance. In accordance with this definition, if we take two points, p and q (fig. 19) in the Cu and Zn respectively, the whole electromotive force will be V p - V, + E zc and the current will be given by

Vp-V γ + E zc = RC (10), where R is the whole resistance of pq. N p - V i is sometimes called the “external,” and E zc the “internal” electromotive force. If p, q include more than one contact of heterogeneous metals, we have only to add on the lefthand side of (10) the corresponding internal electromotive force for each discontinuity.

If p and q be connected by wires of the same metal, say copper, to the electrodes of a Thomson’s electrometer, then the electrometer will indicate a potential difference, Vp - Υ q + E zc , and not V p - V, as might at first sight be suspected. No electricity can flow through the electrometer, hence the copper wire attached at p, and the pair of quadrants to which it leads (we may suppose the quadrants made of copper, but in reality it does not matter, see below, Origin of Electromotive Force), will be at potential V p . But owing to the contact force between the Zn and Cu at q, the wire from q and the quadrant to which it leads will be at potential V σ - E zc . It appears, therefore, that the electrometer indication corresponds to the whole electromotive force between p and q, and is proportional to the whole resistance between p and q, no matter what metals the circuit may include. This conclusion was verified by Kohlrausch. His method rested on the principle of Volta’s condensing electroscope.

lie used an accumulator consisting of a fixed plate B, and an equal movable plate A, which could be lowered to a very small fixed distance from B, and raised to a considerable distance, so as to touch a fixed wire leading to a Dellmann’s electrometer. The plate A was lowered and connected with p, while q and the fixed plate were connected with the ground; the connection with p was then removed, and A raised, its potential thereby greatly increasing owing to its greatly diminished capacity. This increased potential was measured by the electrometer, with which A was in connection through the fixed wire. In one of Kohlrausch's experiments, he found for the electromotive force between a fixed point of the metallic circuit and four points, such that the resistance between each adjacent pair was very nearly equal, the values 0'85, 1’81, 2 - 69, 370; the values calculated by Ohm's law were 0 - 93, 1'86, 2'80, 373. He also examined the fluid part of the circuit, and still found a good agreement between theory and experiment. (See Wiedemann, § 102.)

The laws of current distribution in a network of linear circuits were first studied by Kirchhoff. He laid down two general principles which are very convenient in practical calculations.

I. The algebraical sum of all the currents flowing from any node of the network is zero. *

II. If we go round any circuit of the network, then no matter how many meshes it may include, or what conductors may branch off at different parts, we have

E = R 1 C 1 + R 2 C 2 + · · · · ‰Cn , where E is the whole internal electromotive force, and R 1 , R 2 . . . . C 1 , C 2 . . . are the resistances and current strengths in the different parts of the circuit.

The first of these principles is simply the law of continuity, and the second is got at once by applying equation (10).

We give here an investigation of the currents and potentials in a network of conductors. The method and notation are taken from Maxwell, vol. i. § 280. Let A 1 , A 2 , ... A n he n points, connected by a network of ⅛n(n - 1) conductors (that being the number of different pairs of conductors that can be selected from the n). Let C p ,, E p ,, K m denote the current strength, internal electromotive force, and conductivity, i.e., the reciprocal of the resistance, for the conductor A p A.. Let, moreover, the potential at A p be P p , and the current of electricity which enters the system there be Q p . It is obvious from our definitions of the symbols that

K,>, = K p9 , C m = - C jp, E pi = - E tp , and, by the condition of continuity, that

Qι + Q s + .... +Qn — 0.

At the point A p we have tpi + Cp j + .... + C pn = Q p . . . (a). Now

C p , — K p ,( I , p - P, + E p ,) .... (β).

Hence (a) becomeβ

I⅛(P 1 - P p ) + K p2 (P 2 - P f ) +. . . . + K pn (P n - P p )

= K p ιE p ι + . . . . + Kp n E pn - Q f . . . (γ).

The symbol K pp does not occur in this equation, and has no meaning as yet. Let us define it to mean - (Kp j + K p2 .... K pn ), where K pp does not occur. Then we have k p ι + K p 2 + . . . + K +∙.∙ + K pn = 0, . · · (δ) and, multiplying by Γ p - P r ,

K pl (P p - P r ) + . . . + Kp p (Pp - P r ) . . . Kp, l (P p - P r ) = 0 . Adding this last equation to (γ) we get

K p ι(P 1 -Pr) + Kp s (P 2 -P r ) + . . . + Kp,,(P n -P r )

— IVpiEpi + . . . + K pn E pn - Qp . . (∈).

In this equation the term whose coefficient is K pr of course vanishes. By giving p all possible values except r, we get a set of n - 1 equations to determine the n -1 quantities P, - P r , P 2 - P r , &c. Hence if M rr denote the minor of K 1 ∙r in the determinant ∆ = (FrχK 22 .... K, ln )j and if M rrp denote the minor of K f9 in M rr , we have

( fp - f*r)Mrr — { K χ 1 E, j + Kj ,E 1 2 + ... + Kχ n El l , — Q 1 } M r rlp + {K 21 E 21 + K 22 E 22 + a -QjMrr⅛

+ &c {ζ h where of course E 11 and E 22 are zero, and M rrrp does not occur. 1 it is expression is linear in the letters E and Q, and the principle of superposition holds, as we saw it ought to do in all applications of Ohm’s law.

Consider the particular case in which all the Qs and Es vanish, except E⅛ and E b j ( = — E⅛), we then have the case of a linear circuit in which an electromotive force E iw is introduced into A 1 A w . We get from (f)

Γ,-Γ.= ⅞⅛ l JU,-M,~ p ⅛. and γ, ~ p '= i ⅛'< m →- m "->∙

Hence

Tl∙ - I∖ - J⅛f rr (Mrrlp - Mpr⅛ - M r rrn p + M rr ∙,) , and p — ∕⅜t 1Γ tr p 1 Mr, (51 rr⅛> - aln√ 2 - M rr ,Hp +M rrm ,) ... (η' f .

Similarly, if C fa , be the current in A i A m due to an electromotive force L m in A w , we get r K⅛d⅛E pg z .,

~ M— ^^ ^lrrpw - M∏∙oi + M r rom) . . . . (θ) .

Now, since Δ is a symmetrical determinant, M rr √ p = M rr pj, &e., and the expressions within brackets in (η) and (0) are identical. Hence follows the important proposition :—

If au electromotive force equal to unity, acting in any conductor A i A m of a linear system, cause a current C to flow in the conductor A p A,, then an electromotive force equal to unity, acting in A r A f , will cause an equal current C to flow in A j A m .

If we suppose all the conductors of the system except AjA m and ApA, removed, and AjA p and A m A, joined by two wires, in such a way that for electromotive force unity in AιA∣> the current in A p A, is C then the conductivity of the circuit which we have thus constructed would be

I^ZB∣hp<y

^^M^r ' ^ rr ∙P i - M r rpn - δI r rQ( + M rrjw ); this might be called the reduced conductivity of the system with respect to A p A q and A∕Λ m . V hen the expression within brackets vanishes, the conductors A p A q and AjA m are said to be conjugate. The reduced resistance in this case is infinite, and no electromotive force in A∣A m , however great, will produce any current in A p A i , and reciprocally.

Similarly, we may prove that if unit current enter a linear system at A; and leave it at A m , the difference of potential thereby caused between A p and A, is the same as that caused between A l and A m , when unit current enters at A p and leaves at A q . (See Maxwell.)

The case of several wires forming a multiple arc very often occurs in practice. r

Let AB, CD (fig. 20) ∖

be two parts of a circuit * R3 ù c A lj whose resistances are R ∖ “χ

and S, and let the cir- —'"

cuit branch out between f ⅛∙ 2 θ∙

B and C into three branches of resistances R 1 , R 2 , R 8 .

We have Yd - Vc = R 1 C 1 = R 2 C 2 = R 3 C 3 , and

⅛ c

. θι - j 1 i » C i — &c.

R 1 + iξ + iξ

Also Va-Vd = V a -V b +v b -Vc + V c - (R + p + S)C, whe’

l,l + ± + A.

y R 1 + R 2 + R a

Hence current in each branch is inversely proportional to the resistance, that is directly proportional to the conductivity; and the reduced conductivity of the multiple arc is equal to the sum of the conductivities of its branches. These statements are obviously true for any number of branches.

Some of the most important applications of the theory of linear circuits occur in the methods for comparing resistances. The earliest method for doing this consisted simply in putting the two conductors, whose resistance it was required to compare, into a circuit which remained otherwise invariable; if the current, as measured by a galvanometer, was the same, whichever conductor was in the gap, it was concluded that their resistances were equal. The difficulty in this method is that the electromotive force and internal resistance of the battery are supposed to remain constant, a condition which it is excessively hard to fulfil.

This difficulty can be avoided by using a differential galvanometer, or the arrangement of conductors called Wheatstone’s bridge. The differential galvanometer differs frum an ordinary one simply in having two wires wound side by side instead of a single wire. If we pass equal currents in opposite directions through the two wires, the action on the needle is zero, provided the instrument be perfectly constructed. If the currents are unequal, the indication will be proportional to the difference of the current strength.

If the coils are not perfectly symmetrical, but such that the deflection due to a current c in one is mc, and in the other ne, where m and n are the “constants” of the two coils, then the deflection for currents c 1 and c 2 is mc 1 - ne 2 .

Fig. 21 gives a scheme of the arrangement for measuring resistances with this instrument. V is the battery inserted in the common branch ED of the two circuits, which convey currents dividing off at D, and going in opposite directions round the coils of G. If we wish to measure the resistance of a wire, it is inserted at AB by means of binding screws or mercury cups, and the resistance of the other circuit is varied until there is no deflection; then AB is replaced by a known resistance, which is made up until there is zero deflection as before.

It is obvious that the only requisite here is that the resistances of EFK, EA, BL, and the galvanometer coils should remain constant. Variations in the electromotive force or internal resistance of the battery dc not affect the result.

The method which we have thus sketched is the best way of using the differential galvanometer, and it does not matter even if the coils are not exactly symmetrical. Let the constants of the coils M and N be m and n, so that the deflection due to currents c 1 and c 2 in M and N is me, - nc a . Let the resistance from E to D in the single branch be B, and in the circuits EFK and EABL, which pass round M and N respectively, R and S + U, U being the resistance between A and B, which is such that the deflection is zero. Then

0 = rnc 1 -ne i ≈ w(S + U) - nR ... (a), where E is the electromotive force of the battery, and

D = (R + S + U)B + R(S + U).

Suppose we substitute U' for U, and arrange U ’ so that we have again zero deflection. Then

0= ∣m(S + U')-nR∣ J (5).

From a and β we get U = U ’.

For farther details concerning this method, see Maxwell, vol. i. § 346, and Schwendler, Phil. Mag., 1867.

The differential galvanometer method was much used by Becquerel and others, but it is now entirely superseded as a practical method in this country by the Wheatstone’s bridge method. Suppose we have a circuit ABDC of four conductors. Insert a galvanometer G between B and C, and a battery between A and D. Adjust say the resistance AB until the galvanometer in BC indicates no current. The bridge is then said to be balanced, and the potentials at B and C must be equal. But the whole fall of potential from A to D along ABD is the same as that along ACD; hence if the fall from A to B is to be equal to that from A to C, we must have

R T S”U’

where R,S,T,U are the resistances in AB, BD, CA, DC. This is the condition that BC and AD be conjugate. We might have deduced it as a particular case of the general theory given above. Hence if we know the resistances S,T,U, we

ST get in termsof these R = —. S is often called the standard resistance, and T, U the arms of the bridge or balance. The sensibility of this arrangement may be found practically by increasing or decreasing R so as to derange the balance. The largest increase which we can introduce without producing an observable galvanometer deflexion measures the sensibility of the bridge.

If we had a given set of four conductors, and a battery and galvanometer of given resistance, then it may be shown (see Maxwell, vol. i. § 348) that the best arrangement is that in which the battery or galvanometer connects the junction of the two greatest resistances with that of the two least, according as the former or the latter has the greater resistance. The practical problem might take another form. We might have given a resistance, and have at our disposal known resistances of any desired magnitude to form our bridge. We might also suppose further that we had given the total area of the plates of our battery, and the dimensions of the channel in which the galvanometer wire was to be wound. We may neglect the thickness of the silk coating, or assume that it is proportional to the thickness of the wire.

Then, B and G being the resistances of the battery and galvanometer, the electromotive force E α√B, and the number of turns in the galvanometer ocV , G.

Let us put S≈=yR, T = zR, andU = yzR. These resistances would balance; let us however put (1 +ar)R in the branch AB instead of R, the others being unchanged, and calculate the effect on the galvanometer in G, which we put proportional to the current in BC’, and to the number of turns on galvanometer. Then, from equation (η) (or Maxwell, vol. i. 349), we find that the deflection 8 varies as yz√BG

(1 + y)(l + Z)BG + y(l + z) 2 BR + z(l + y) 2 GR + yz{ 1 + y)(l + z)R* ’ in order that δ may be a maximum, we must have

G {(1 + y)(l+z)B+z(l + r∕) 2 R} = y(l + z) 2 BR + yz( 1 + y)(l + z)R*∙ (a). B{(1 + 2 ∕)(1 + z)G + 2 ∕(1 + z) 2 R} = z(l + y) 2 GR + yz(l + j∕×l + z)R 2 (3), BG=zR 1 .... (γ ),

BG=yR ’. (8).

a an<l β give at once by addition and subtraction

S-⅛T⅛ “d bg =^ r, ∙

or B=z⅛ . . . (e),

1+z

°-≈'f⅛ lt · · · ∙ f 0∙

Combining the four equations (γ), (δ), (e), (f), we get y = z = landB-G-R = STU.

It appears, therefore, that when all the resistances on the bridge are at our disposal, we ought to make them all equal to the resistance to be measured, or come as near this as we can; e.y., if we had a very small resistance to measure, we should make the arms of the bridge small, and take a small-resistance in preference to a high-resistance galvanometer.

In order to carry out measurements of resistance with ease we must possess a series of graduated resistances, with which we can compare any unknown resistance, and of which we can make the arms of our balance, Ac. Again, if the measurements of one electrician are to be of any use to another, there must be a common standard. It would be most convenient to have only one standard for all nations, and this standard might be either arbitrary, like the standard of length, or absolute in some sense such as we have defined above. Arbitrary standards have at different times been proposed by Jacobi and others. The mercury standard of Siemens, to which we alluded in the historical sketch, has obtained great prevalence on the Continent. The British Association unit or ohm is an absolute unit, inasmuch as it professes to represent in electromagnetic measure a velocity of 10 9 centimetres per second, or, taking the original definition of a metre, an earth quadrant per second. It happens, by a curious accident, that the mercury unit and the ohm are very nearly equal, the latter being expressed in terms of the former (according to Dehms and Hermann Siemens; see Wiedemann, Bd. ii. 2, § 1074) by the number 1Ό493.

One of the earliest instruments for furnishiug a graduated resistance was the rheostat, brought into use by Wheatstone, but also invented independently by Jacobs at St Petersburg about 1840.

It consisted of two cylinders of equal diameter, one of wood and one of brass. A wire, whose extremities were in connection with the metallic axes of the cylinders, was wound in opposite directions round the cylinders. The axes of the cylinders were connected with two binding screws by means of sliding contacts. The part of the wire which does not lie on the metal cylinder is the only part that produces resistance between the binding screws; and, by winding and unwinding, we can increase or diminish the resistance continuously to a known extent, means being provided for measuring the angular rotation of the metal cylinder.

We shall not stop to consider the defects of this instrument, which is now never used for delicate work. Its place is taken by resistance boxes, containing coils of wire whose resistances are different multiples of the unit of resistance (in this couutry always the ohm). The reader will find a full account of the methods by which the standards are reproduced in the collected reports of the Committee on Electrical Standards. The usual material for the wire of resistance coils is German silver. Most of the copies of the ohm issued by the British Association were made of an alloy of two parts of silver to one of ρlatiuum. The great advantage of alloys is that the variation of resistance with temperature is small for them; in the PtAg alloy, for instance, it is less than a tenth of the value for an average pure metal. To secure insulation the wires are carefully coated with silk, and after winding the coil is immersed in melted paraffin. To get rid of electromagnetic and inductive effects, the wire on resistance-coils is doubled on itself before being wound, so that, when a current passes through the coil, there are always two equal and opposite currents at each point. The terminals are formed by stout pieces of copper rod, whose resistance is either included in the coil, or is so small that it may be neglected. The connections for small resistances are managed by means of mercury cups, with pieces of amalgamated copper at the bottom, on which the copper electrodes are made to press.

For ordinary purposes the coils are arranged in a box (fig. 23), the terminals being stout pieces of brass fixed on the ebonite lid; conical brass plugs inserted between these pieces serve to throw the coils in and out of circuit. The box represented in fig. 23 is specially arranged for use in Wheatstone’s bridge. Iu E,F,Gwo have a series of coils, 1000, 100, 10, 10, 100, 10θ0; these are used for the arms of the bridge. Iu A,C,D there are sixteen coils, 1, 2, 2, δ,, 10, 20, 20, 50, &c., which give us any resistance of a whole number of ohms from 1 up to 10,000. In actual use the resistance to be measured is inserted between A and G, D and E are connected by α stout piece of copper, the galvanometer is inserted between F and

A, and the battery between E and G. The resistances of the arms of the bridge are taken equal, and as near the resistance to be measured as possible. In this way the resistance of any conductor may be very quickly found to an ohm. If it is desired to go farther, we may proceed thus. Suppose that we have found that a resistance lies between 5 and 6, put in the arm FE 100, and in FG 10, let the resistance in DCA, when there is a balance, be 57, then the resistance of the conductor is - 1 ⅛⅞ × 57, or 5 , 7. Similarly we might go to a second place of decimals by putting 1000 in FE and 10 in FG. There is a limit, however, to this process, because the increase in the resistance of the arm decreases the “sensibility’’ of the bridge. Another method is to balance as nearly as possible, and then interpolate by taking the deflection of the galvanometer. Suppose, for instance, in the above case, that, with 5 ohms in DCA, the deflection was 21 in one direction, and, with 6 ohms, 9 in the other direction, then, taking the deflection proportional to the deviation from balance (see formula for δ above), we have

. 21 resistance = 5 + —. 1 = 5 ’ 7.

We might also construct small graduated resistances; and this would enable us to use smaller arms in the bridge, and thus increase the “sensibility” when used to measure small resistances. Owing to the multiplication of connections, there is a limit to the ordinary resistance box arrangement. The difficulty may be evaded to a certain extent by using conductivity boxes, according to Sir W. Thomson’s suggestion, where the resistances are arranged abreast, so that a small alteration of the resistance is brought about by adding on a very great resistance to the multiple arc. The rheostat principle has been used by Poggendorff in his rheocord for producing small resistances. He stretches two platinum wires side by side; on these is strung a hollow box filled with mercury, whose longitudinal motion is read off on a scale. If this arrangement be thrown into any circuit by means of two binding screws connected with adjacent terminals of the wires, the parts of the two wires up to the bridge give a small resistance, which may be adjusted at pleasure.

In the quicksilver agometer of Müller (Wiedemann, i. § 160), the resistance is formed by a column of mercury of variable length. We may remark here that difficulties equally arise in constructing very large resistances. To get such within reasonable compass the wire must be ex ceedingly thin and the insulation very good. Messrs Warden and Muirhead have wound coils of fine wire, giving a resistance of 100,000, and have constructed in compact form resistance boxes up to 1,000,000, or a megohm, and beyond. They have also given practical form to a suggestion of Phillips to utilize the resistance of carbon, by drawing fine pencil lines on ebonite or glass; they mix plumbago with the pulp in the ordinary process of paper manufacture, and thus produce a species of carbon paper. A strip of this about 21 in. long and ’ 5 in. broad gives a resistance of about 50,000. This seems a valuable invention; but we are not aware how far it has stood the test of practical use.

Selenium and tellurium have been proposed as material for high resistances, but owing to the variability of their resistance under the action of light, Ac., they are unfit for the purpose.

The best method for comparing resistances with great accuracy is the modification of Wheatstone’s bridge introduced by Kirchhoff (fig. 24).

KL is a platinum-iridium wire, DK and HL are stout copper terminals to which it is soldered, DAE, EGF, FBH are stout copper pieces with binding screws and terminals for mercury cups, by means of which resistances R,T,U,S can be inserted at D,E,F,H. A, B, and G are binding screws for the battery wires and one end of the galvanometer wire. The other end of the galvanometer wire is screwed to a spring contact piece fixed to a sliding block at P; when the button of this block is depressed, contact is made with KL, at a spot which is definite to an eighth or tenth of a millimetre. Platinum iridium is chosen for KL, because it is hard and tough, not liable to be scratched or abraded by the contact piece, does not oxidize or amalgamate with mercury, and changes very slightly in resistance when the temperature alters. The wire must be calibrated to find what correction, if any, must be applied for variation of resistance per unit of length at different parts; for methods of doing this see Matthiessen and Hockin; Brit. Assoc. Reports on Electrical Standards, p. 117; or Foster, Journ. of Society of Telegraphic Engineers, 1874.

Kirchhoff’s arrangement may be used in the ordinary way after we have made special experiments to determine the resistance of the connections, &c. Professor Foster (Z.c.) has given a very useful method, by which the difference of two resistances can be got independently of the resistances of the connections. Suppose we wish to find the difference between R and S, which we suppose so near each other that, with the arms T and U approximately equal, there will be a balance when P is somewhere on KL. Let the reading for the position of the block be x, taken from left to right. Interchange R and S, balance again, and let the new reading be x' (we suppose the difference between R and S so small that P is still on KL); then, if p. be the resistance of unit length of KL, R - S = p(z' - x).

For, if a represent the resistance of the connections in DK, β the same for the other end of the wire, and if T and U include the resistance of the invariable connections, then we have

R + a + ux T

S + 0+M(Z→β)~U ’

where l=length of KL. Hence

R + α + μX _ T

R + S + a+ß + jtiZ 1 +U

Similarly

S ÷ α + μx' T

R + S + a + β + μ Z “TV U ’ therefore

R-S=μ(ar'-x).

If we have to find the resistance of a thick cylindrical body, what is really wanted is the ratio of the current strength to the difference of potential between the two ends, when the current flows parallel to the axis at every point. The last condition is not generally fulfilled. It is obviously not so in the case where the cylinder is joined up with a thin wire. In cases where we wish to compare the specific resistance of two metals which we possess in cylindrical pieces, we get over the difficulty by observing the potential at a point at some distance from the end of the piece, where the flux is parallel to the axis at all points of the section.

Matthiessen and Hockin used the following method for this purpose (fig. 25). Tfie two pieces XZ, YZ are soldered together and connected in circuit with two resistance coils A and C, and a graduated wire PR as before. S, S' are two sharp edges, at a measured distance apart, fixed in a piece of ebonite or hard dry wood, and connected with mercury cups. T, T' is a similar arrangement for YZ. The galvanometer is inserted between S and Q, and the position of Q is found for balance; then the terminal is shifted to S', and if necessary the resistances A and C altered, so as to keep their sum constant, until balance is again found. The same is done for T and Τ'. Then, XS denoting the resistance between X and S, and Λ ιs C 1 the values of A and C in the first case, and so on, we have

XS A 1 + PQ 1 XS'-A 2 + PQ 3 XY - R ’ XY R ’ where

R=A 1 + C 1 + PR = A 2 + C s + PR

Hence

SS' - A a -A 1 + Q 1 Q a XY~ R

Similarly

TT'_ A 4 — A 3 + Q3Q4 XY - R ·

Therefore

SS' _ A 2 - A 1 + Q 1 Q a TT A 4 - A 3 + Q 3 Q 4

This gives us the ratio of the resistances between SS' and TT w . The nlethod does not depend for its success on the goodness of the contacts at SS', &c. Another ingenious arrangement for effecting a similar purpose is due to Thomson, and will be found described , in Maxwell, vol. i. § 351.

In measuring very large resistances, such as the insulation resistance of a telegraph cable, it is convenient to use the quadrant electrometer. One end of the cable is connected with one electrode of a condenser, the other end of tbe cable is insulated, and the other electrode of the condenser put to earth. The condenser is charged, and the difference of potential between its electrodes measured by means of the electrometer. If E 1 , E s be the value of the difference at the beginning and end of · an interval of t seconds, and if S be the capacity of the condenser in electromagnetic measure, then the resistance of the cable is t S(log f E 1 ~ log,E 2 )

in electromagnetic measure. If the condenser itself leaks, we must determine its resistance by insulating the electrodes and operating as before. Then, regarding the circuit in the first experiment as a multiple arc, composed of the insulation of cable and the dielectric of condenser, the true conductivity of the cable envelope is the difference of the conductivities obtained in the two cases. Several other methods might be used to compare metallic resisistance but they are of small importance compared with those we have now been describing.

The reader who desires information concerning the application of Ohm’s law to conductors other than linear will find the sources sufficiently indicated in Wiedemann’s Galvanismus; some of them have been alluded to in the Historical Sketch.

Application of Ohm ’ s Law to Electrolytes.

In our discussion of Ohm’s law, we have hitherto had in view principally the metallic part of the voltaic circuit. We now turn our attention more particularly to the fluid parts. It is of no importance in the present connection whether the fluid forms part of the “battery” or “electromotor,” or whether it is inserted outside the battery; the only difference in these two cases is, as we shall hereafter see, that in the former case energy is being absorbed by the current, and in the latter it is being evolved. In many respects the properties of the metallic and fluid parts of the circuit are alike: the electromagnetic action is the same for both; heat is also developed in the body of the conductor, whether metallic or fluid, according to the same law. But there is one peculiarity about a large class of fluids which has no analogue in purely metallic conduction, viz., that in them the passage of a steady current of electricity is invariably accompanied by chemical decomposition, definite in kind and quantity. To such* fluid substances Faraday gave the name of electrolytes.

For example, suppose we fill a small beaker with a solution of zinc chloride (ZnCL>), and suspend in the liquid two strips of platinum foil (called electrodes'), at a moderate distance apart. Let a current enter at one of these strips, which we shall call the anode, and leave at the other, which we snail call the cathode. It will be found that the solution continues to decompose so long as the current passes, zinc appearing at the cathode, and chlorine at the anode. The metallic zinc precipitates, and the chlorine combines with the platinum of the anode to form ρlatiuic chloride.

It is obviously essential in an electrolyte that it should be a compound in some sense or other. It is not, however, true that all compound bodies are electrolytes. Fluidity is also a necessary condition, whether attained by heating to the melting-point, or by dissolving in water or other solvent. Faraday established as a law, to which there appear to be few, if indeed any, exceptions, —that all substances which in the solid state are very bad conductors, but conduct on being heated to the melting-point, are electrolytes, i.e., are decomposed by the passage of the electric current. Faraday thought that periodide of mercury, fluoride of lead, and some other bodies were exceptions to this law; but later researches seem to have established that this is not so. (Cf. Experimental Researches, 414, 439, 1340, <fcc., and Wiedemann’s Galvanismus, i. § 191, <fcc.) The conductivity of electrolytes in solution also increases rather quickly with increase of temperature, while the conductivity of metallic conductors, on the other hand, diminishes, but more slowly, as the temperature rises.

In considering the passage of the current through electrolytes, it is convenient to distinguish two cases. First, let there be a steady, or at least permanent current, and a continuous evolution of the products of electrolytic decomposition (these are called the “zone,” anion and cation at the anode and cathode respectively). The amount of ion that appears at an electrode in a second is equal to the strength of the current (supposed constant during a second') multiplied by a constaid called the electrochemical equivalent of the ion

The electrochemical equivalent is proportional to the chemical equivalent, account being taken of the “valency” of the ion. (See art. Electrolysis.)

. For instance, if C be the strength of the current in the illustrative case above, then the amount of zinc deposited at the cathode in time t will be zCt, and the amount of chlorine liberated at the anode cCt, where z and c the electrochemical equivalents of zinc and

., . , 65 ...

entonne, and z: c :: 35’5, zinc being divalent. If a cell con taining .ea∣l chloride (PbCl a ) were also inserted into the circuit, the same amount of chlorine would be liberated at the anode, and the amount of lead precipitated at the cathode would be pCi, where 207 65

p: z: c :: ~2*: ο: 35'5, τ.e. :: 103 ’ 5: 32 5: 35 5

As the electrochemical decomposition (“electrolysis”) goes on, the surface of the electrodes is altered. Tn some cases the ion is merely deposited on the electrode, in other cases it combines more or less intimately therewith; but in general there is an alteration of the nature of the contact, and a consequent alteration of the electromotive force at the surface of the electrode. Experiment shows that this electromotive force, in a great many cases, tends to oppose the passage of the current. So that if we insert an electrolyte into any circuit, the current starts with a certain value, and falls more or less quickly, until it reaches a limit at which it remains steady. The opposing electromotive force of “polarization,” as it is called, has then reached its maximum, and the deposition of the ions goes on without further alteration of the contact surfaces. It is obvious that this limit may be reached under a variety of different circumstances (vide infra, p. 86). There is also another phenomenon, the possibility of which we must not overlook, viz., an alteration of resistance, owing to the presence of the ions at the electrodes. This resistance, due to the ions, has been called the “transition resistance.” The enfeebling of the current by the electromotive force of polarization might, as far as the observed result is concerned, be due entirely to an increase of resistance, or to a transition resistance, and such was the explanation given by the earlier physicists. It is easy, however, to show that there is an actual electromotive force of polarization; for, if we disengage our electrolytic cell from the battery, and connect its electrodes with a galvanometer, a current is indicated, which passes through the cell in the opposite direction to the original current. This could not be due to any transition resistance, but must arise from an opposing electro motive force generated by the passage of the battery current. This point can be illustrated by a hydrodynamical analogy. If we attempt to force water through a narrow capillary tube, or through a wide vertical tube against gravity, there is an opposing force in both cases. But, when we remove the pressure, the water has a tendency to return in the latter case, but none in the former. The former case represents a transition resistance, the latter an electromotive force of polarization.

Without denying the existence of a transition resistance, we see that an electromotive force of polarization actually exists. In some cases, e.g., amalgamated zinc in zinc sulphate, it is very small; in other cases, e..g., platinum electrodes in dilute sulphuric acid, it may considerably exceed the electromotive force of a Daniell’s element.

We have, up to this point, been treating the case where a permanent current finally flows through the electrolyte; but there are cases where the existence of such a current would violate the principle of the conservation of energy.

Suppose that a single Daniell’s cell is the electromotor, then (see below, p.90) if a current C is sent for a time t, an amount of energy dCt is absorbed in the cell, d being constant. Suppose, farther, that the excess of the intrinsic energy of the ions, in the state in which they are being delivered in the electrolytic cell, over that which they possess when in combination is w, then if a current C pass for a time t, an amount of energy wCt will be evolved. But if j∕'>cf, this cannot go on for any time however short, no matter how feeble the current may be, otherwise more energy would be evolved in the cell than is absorbed in the battery.

If we insert an electrolytic cell containing dilute sulphuric acid along with a galvanometer into a circuit in which there is a single cell of Daniell, we observe the galvanometer needle swing out vigorously, and then settle down to a small and gradually decreasing deflection. The current ultimately becomes zero; but the time it takes to do so may be considerable, and varies with the nature of the electrodes. If we remove the battery after the current has stopped, and connect the polarized cell with the galvanometer, we observe an initial swing very nearly equal to the former but in the opposite direction, and a corresponding deflection, which after a time disappears entirely. Although, as a rule, a sensible time elapses before the polarization reaches its maximum, yet it is important to remark that it may rise to a very considerable fraction of the maximum in a very short*time indeed. Edlund found that in a certain case the electromotive force of polarization reached 0 - 57 of a Daniell in about - 5 1 σ of a second. Bernstein has recently arrived at results of a similar kind. He found, for instance, that platinum plates, polarized to 1 , 85 of a Daniell, fell, when the resistance of the circuit was 7 - 46 Siemens units, to 1 , 57 in Ό0111 sec. This rapidity of the rise and fall of the polarization is of very great importance, and has, we think, been overlooked by some experimenters.

In cases where the polarization does not reach its maximum, no liberation of gas or other ion is observed, such as is seen with a permanent current, and it might of course be denied that chemical decomposition takes place at all. We shall, however, assume that Faraday’s law holds for this case also, and assert that the current in the first instance actually passes through the liquid and produces chemical decomposition, according to the same law as a permanent current, and that this goes on until the accumulation of the ions has generated an opposing electromotive force, equal to that of the battery, when of course the current must stop. We cannot justify this position very easily by direct experiment; yet there are many facts to support it, and so long as it is tenable it seems to afford the most philosophical view of the matter.

Having explained the phenomena of polarization so far as is necessary for our immediate purpose, we now proceed to inquire how far experience justifies the application of Ohm’s law to electrolytes, or, which is much the same thing, to examine how far the methods of different physicists for measuring electrolytic resistance have led to concordant results.

One of the earliest methods, in which polarization was eliminated, was that of Horsford. He filled a rectangular trough with the electrolyte, and inserted in the trough two electrodes very nearly fitting the cross section. These electrodes could be set at different measured distances apart. They were coated on the further side with nonconducting substance, so that the current could flow between the opposed sides only. In this way he secured that the stream lines in the neighbourhood of the electrodes should depend as little as possible on the distance between them. This trough was inserted in the battery circuit along with a tangent galvanometer; then the distance between the plates was decreased, and a metallic resistance R inserted in the circuit, so as to bring the current to the same strength as before. The current being the same in both cases, it is assumed that the polarization in both is the same, in which case the resistance of a length of the electrolyte equal to the difference of the distances between the electrodes in the two cases is equal to R. Knowing the section of the trough, we might calculate from R the specific resistance of the electrolyte. If the values arrived at be the same when deduced from different lengths of the electrolyte, and for different strengths of current, it may be concluded that Ohm’s law applies. The application of this method requires the passage of a permanent current, in consequence of which the ions appear at the electrodes, and the solution in the neighbourhood becomes altered; so that it is difficult to make certain that the polarization is exactly the same in the two cases, and that no resistance of transition is generated. Matters may be mended a little by passing the current for the same time in both cases; but this is scarcely a satisfactory remedy. Still valuable results were obtained with this method by Horsford and Wiedemann; the latter, in applying it to silver and copper solutions used electrodes of silver and copper respectively, whereby the polarization to be eliminated was very much reduced. .

Taking advantage of the discovery of Matteucci and Du Bois Reymond, that carefully amalgamated zinc electrodes in a neutral solution of ziuc sulphate are not polarizable, Beetz determined, by means of Wheatstone’s bridge, the resistance of various solutions of this electrolyte.

The liquid was inclosed in a cylindrical tube, 29 - 7 cm. long, with a mean section of l - 4051 sq. cm. Amalgamated zinc plates were applied to the ends of the tube, and fastened on by india-rubber collars. The ends were then inserted tightly into openings in the sides of two bottles which were filled with the solution (the same as that contained in the tube). The thick electrodes leading to the discs, and the backs of the zinc discs themselves, were lacquered, to insulate them from the liquid in the bottles. The whole apparatus was immersed in a trough of water, which could be heated to any desired temperature.

In the course of his experiments Beetz demonstrated the absence of polarization when amalgamated zinc electrodes are used, and eliminated the transition resistance by boiling the electrodes in zinc sulphate, and transferring them to the ends of the tube without exposure to the air.

Beetz farther proposed to find the specific conductivity of other electrolytes in terms of that ef zinc sulphate, by experimenting on closed circuits consisting entirely of the electrolyte to be examined. He tried damping experiments for this purpose, but the effects to be observed turned out too small for accurate observation.

Paalzow inclosed the electrolyte to be examined in a siphon, the two ends of which dipped into vessels of porous clay also filled with the electrolyte. The clay vessels were immersed in beakers filled with zinc sulphate, at the bottoms of which were placed large amalgamated zinc discs, which formed the electrodes. The only polarization or transition resistance to be feared is that at the boundary of the two liquids, and this is very small. What little remained was eliminated, as in Horsford’s method, by taking differences.

The resistance of the whole arrangement was measured by means of Wheatstone’s bridge, and then the siphon was replaced by a shorter one filled with the same liquid. If R 1 , R 2 be the resistances found in the two cases, R 1 - R s is obviously the resistance of a length of the electrolyte equal to the difference between the lengths of the siphons. If R 1 ’, R 2 ' be similar values obtained when the electrolyte is replaced by mercury, then the specific resistance of the R — R

electrolyte is —A——A , that of mercury being taken as unity.

K 1 — R 2

The most important of all the recent researches on the application of Ohm’s law to electrolytes are those of F. Kohlrausch and Nippoldt. In order to avoid the effects of polarization, they used the alternating currents of an electromagnetic machine. These currents varied very nearly as the sine of the angle of rotation, and could be sent in rapid succession through the electrolyte. The whole quantity of electricity that passes in the first part of any alternation is exactly equal and opposite to that which passes in the second; hence equal quantities of the two ions (say H and 0) will be separated at each electrode. If the H 2 and 0 combine to form water, it is obvious that, on the whole, there will be no resultant electromotive force of polarization either way; and if they coexist side by side without combining, there will still be no resultant electromotive force, provided the electrodes be exactly similar. There are two advantages in this method. There is no evolution of gas or other ion, and consequently no alteration of the solution and electrode, such as goes on with a constant current. We have, besides, another great advantage, which is denied us with constant currents,— viz., that by increasing the size of the electrode, we can diminish the effects of polarization.

The whole amount of electricity which passes in each induction current is the same, and consequently the whole amount of ion deposited on the electrode is the same; hence, if we increase the surface of the electrode, the density of the deposit is decreased in an inverse ratio. Now, the researches of Kohlrausch and Nippoldt have shown that, within certain limits, the electromotive force is proportional to the surface density of the deposit. Hence, by sufficiently increasing the surface of the electrodes, the polarization may be made as small as we please.

In the earlier experiments platinum electrodes, having a surface of 1Ό8 cm. were used, and it was found that each induction current of the magneto-electric machine deposited on each square millimetre of the positive electrode only c.cm. θf oxygen.

It was therefore expected that the polarization would be insensible, and that the electrolyte would behave like a metallic resistance. The magneto-electric machine and the electrolyte were connected up with an electrodynamometer, and it was found that the deflection of the suspended coil of the electrodynamometer was scarcely sensible when the machine, made 10 revolutions per second, although it was 15 scale divisions when the electrolyte was replaced by 70 Siemens units. On the other hand, when the velocity reached 77 revolutions per second, the deflection was much greater with the electrolyte than with 70 Siemens units. It was found, however, that when the surface of the electrodes was increased to 29 cm. a metallic resistance could be found, which gave the same deflection (w ithin errors of observation) as the electrolyte for speeds varying from 4 , 8 to 76 - 9 revolutions per second.

The above results seem to compel us to one or other of two conclusions,—either that Ohm’s law does not apply to rapidly alternating currents, where the maximum of polarization is not reached, or else that the electromotive force of exceedingly small deposits of the ions must be very considerable. The fact that, under certain conditions, the electrolyte is apparently a better, and under others, apparently a worse conductor than a certain metal wire, seems at first sight rather to point to the former conclusion. On the other hand, the result with the 29 cm. electrodes, is a direct verification of Ohm’s law. Kohlrausch, therefore, adopted the latter conclusion, and justified his doing so by special researches on the electromotive force of small gas deposits. He showed that, with the currents he used, the electromotive force is proportional to the surface density of the deposit, and estimated that the products of decomposition of 7 ⅛ mg. of water per square metre would generate an electromotive force equal to that of a Daniell’s cell. It is of the greatest importance to remark that the polarization effects, from which this result is deduced, must have arisen and disappeared in some cases in much less than τ 1 τr of a second. The anomalous behaviour of the electrolyte with small electrodes is explained by Kohlrausch by taking into account the self-induction of the circuit.

A little consideration will show that the electromotive force due to this cause always opposes the electromotive force of polarization, when the current strength is a simple harmonic function of the time. Let i denote the current strength, reckoned positive in a given direction, then, according to Kolilrausch’s law, the electromotive force of polarization at time t is -pf'idt, where p is the o electromotive force generated by the passage of a unit of electricity; its value depends on the electrolyte and on the electrode being, caeteris paribus, nearly inversely proportional to the surface cf the latter. Let n be the number of revolutions of the machine per second, and let 2τ=T; then we may represent the electromotive force of the machine at time t by — sin — t, and the electromotive force τ τ

of self-induction by where k and q are constants, the latter being the coefficient of self-induction of the circuit (see Electromagnetism, p. 76 ). If w be the whole resistance of the circuit, we may write

J-⅛y>∕‰, or ·

q dP + w di + ^ = p co ≡^'

Neglecting disturbances that die away very soon after starting the machine, we get for the value of », k . τr ~ sin ~t v ∕^ + (⅛-^) 2 where the origin of time has been thrown back by

The deflection α of the dynamometer is proportional to - i i?dt, and may be written ♦

An*

^ 2 "y) i ∙

Kohlrausch found that this formula completely accounted for all the peculiarities in the behaviour of the electrolyte (for the numerical verifications see the papers quoted). We see that the deflection is increased or diminished by the insertion of the electrolyte, according as n is greater or Jess than 7r y∕ and, if n= the insertion of the electrolyte makes no difference. Again, if n = T ∕ — o ’ 2%∖∕ ϊ

the deflection will be the same as if there were no extra current and no polarization. So that, for any given electromagnetic machine, working at any given speed, a certain electrolytic arrangement can be found, which will exactly eliminate the effect of self-induction, and thereby render the efficiency of the machine a maximum. It is obvious too that, with a given electrolytic cell, the deflection reaches a maximum when

P n = -

2%√pj-∣w 2 this maximum was actually observed by Kohlrausch (Z.c.).

Having due regard to these circumstances, Kohlrausch and Nippoldt found that Ohm’s law was applicable to their alternating currents, for electromotive forces varying from over ∣ to under 7 1 f of a Grove’s cell. By using the constant current of an iron-copper thermo-electric pair, they found Ohm’s law applicable to zinc sulphate with amalgamated zinc electrodes, when the electromotive force was reduced to τ 3j- 5 ⅛7 rσ of a Grove’s cell.

It is important to remark that the fact that the electrolyte behaves like a metallic conductor through a considerable range of velocities of the sine inductor, is not a conclusive proof that the last trace of polarization has been eliminated.

In fact, let x be the resistance of the electrolyte, W that of the rest of the circuit, and w the metallic resistance that gives the same electrodynamometer deflection for n revolutions of the inductor per second, then the above formula gives X ~ 2W+ x + w(f q ^ 8⅛ ’) =^W + u¼ - 8⅛ 2 ) ’

since we suppose x very nearly = w. If now p be reduced to a very small value, it may happen, especially for tolerably high speeds, p 2 .

that is very small compared with pq, in which case x-w will be independent of n through a considerable range of speed, and the electrolyte will be replaceable by a wire whose resistance is less than the real resistance of the electrolyte by a small constant quantity.

The earlier results cf Kohlrausch and Nippoldt for sulphuric acid, in which they used 29 cm. electrodes, were affected with an error due to this ca*use, amounting to about 4 per cent. In the later experiments of Kohlrausch and Grotrian, this error was finally eliminated by “pl a-tinizing” the platinum electrodes. Kohlrausch had found that, with “platinized” electrodes of only 1 sq. cm. surface, the polarization of the currents of his sine-inductor was insensible; he therefore concluded that, with 25 sq. cm. platinized electrodes, the residual polarization would be finally eliminated. To make quite certain, he instituted three tests, which were carried out on the method used in all the later experiments on this subject.

The Wheatstone’s bridge arrangement was adopted. Fig. 20 gives a scheme of the arrangement. The fluid and a rheostat occupy two arms of the bridge, the remaining two contain each 100 Siemens units; A is the fixed and B the suspended coil of the electrodyna. mometer, and S the sine-inductor.

In this way, (1) the resistance of a receiver with 25 cm. platinized electrodes was found, when filled first with H 2 SO 4 of maximum conductivity, and secondly, with NaCl, the driving weight of the inductor being varied, so as to give speeds of 10 to 100 revolutions. The results, reduced to a common temperature, were, for the H 2 SO 4 , 141 ’ 73, 141∙64, 141 ’ 52, 141*03, 14L55, and, for the NaCl, ,

366 ’ 27, 366 ’ 23, 366 25, 366 ’ 21 Siemens units, with the driving weights 5, 7 ’ 5,10,15, 20 kgr. respectively. (2) The resistance of a solution of zinc sulphate was found, first, in Beetz’s manner with constant current and amalgamated zinc electrodes; secondly, using alternating currents and the same electrodes as before; thirdly,·with alternating currents and the platinized electrodes; the three results reduced to a common temperature gave 537 - 49, 537*41, 537 ’ 20. The greatest divergence from the mean might have been caused by an error of ∙ 2 ⅛ degree in the temperature measurement. The agreement may therefore be pronounced complete. We think that it must be conceded that the experimental methods just described have solved in a satisfactory manner the problems involved in the determination of electrolytic resistance. We have dwelt on them so long partly because nearly all the information on the subject we possess has been obtained by their means, and partly because they present points of great theoretical interest.

Another method has been employed by Ewing and Macgregor. The electrolyte was inclosed in a narrow tube with wide ends, in which were set platinum electrodes. This arrangement was inserted in a Wheatstone’s bridge, and its resistance measured in the usual way. The precautions against polarization consisted in operating with currents of very short duration, sent through the bridge by means of a “rocker” worked by hand; the resistances in the arms of the bridge were also made large, in order to reduce the rate of polarization as much as possible; another essential feature of the method is the use of a “dead beat” galvanometer with a mirror of very small moment of inertia. The paper of Ewing and Macgregor has formed the subject of a somewhat bitter criticism by Beetz, to which Macgregor has replied.

Battery Resistance.—If the electromotive force and internal resistance of a battery in action were the same, whatever the external resistance, there would be no difficulty in finding the internal resistance by Ohm’s method. We have simply to give two different values to the external resistance, and measure the current in the two cases. The electromotive force does not appear in the ratio of the two current measures; hence, knowing this ratio, we can find the internal resistance. Or we may use an electrometer, and measure the difference of potentials between the two poles of the battery, first, when the external resistance is infinite, secondly, when the external resistance is R. Then, if r be the internal resistance, the ratio of the first electrometer H - f^ τ

reading to the second is —, by Ohm’s law; hence r can be found.

Unfortunately, however, the electromotive force of a battery is not independent of the external resistance. In general, when a battery is circuited through a small resistance, its electromotive force is much smaller than when the external resistance is very great. This arises from the polarization set up by the passage through the battery of its own current, and possibly in some degree from other causes as well. There is also reason to believe that the internal resistance of the battery is a function of the current. This being so, it is clear that a theoretically satisfactory determination of battery resistance cannot be arrived at by such methods as we have described. Since, however, the increase of the electromotive force is very slow after the externa] resistance has reached a certain value, and since the alteration of the internal resistance takes some time, we can get in many cases measurements sufficiently accirate for practical purposes. A variety of methods have been devised with this object, and applied mostly to the so-called constant batteries. It must be remembered, however, that there is something indefinite in the term in ternal resistance, unless the circumstances be given under which it is found. In the method of Von Waltenhofen, the battery is “compensated” by another battery so arranged that no current passes through it; and then this arrangement is slightly altered, so that a very small current passes through the battery. This amounts to finding the internal resistance for very small currents. The method of Beetz also involves the principle of compensation; two batteries are used, but the one whose resistance is to be found is compensator and not compensated. The circuit of the compensator is joined for an instant, and then the compensated battery is thrown in. The assumption in the method is that the electromotive force is the same in the first instant whether the battery is closed through a resistance R or a resistance R'. The results seem to justify the assumption, and to establish the practical value of the method; but there are clearly limits to its application which it would not be very easy to define. Beetz himself shows that the electromotive force of a battery is greater when it is compensated than when it is compensating. A similar objection may be urged agai∩3t the method of Siemens, which again gives good results when properly used. We refer the reader interested in this matter to the sources of information already quoted (see Historical Sketch), and content ourselves with an accountof Mance’s method, which, although subject to the same objection as all the others, is very convenient for rough purposes, and is much employed in this country.

Let A, B, C, D be four resistances arranged in circuit, B being the battery w hose resistance is required. Insert a galvanometer between AB and CD, and a circuit which can be closed and opened by means of a key between AL) and EC. We thus have an ordinary Wheatstone’s bridge, with a key in place of a battery, and a battery in place of the ordinary resistance to be measured. Owing to the presence of the battery, there will be a current through the galvanometer, which will deflect the needle; this deflection is compensated by means of a magnet, and the needle brought back to zero. Then the resistances A, C, D are arranged so that the galvanometer is not affected when the key circuit is opened or closed; when this is so the key and galvanometer circuits are conjugate, and we have AC - BD = 0, from which we can find B, since A, C, D are known. In practice, however, it is impossible in the great majority of cases to fulfil the direction printed in italics. Suppose for a moment we had arranged the resistances so that AC - BD is very nearly but not quite zero, and suppose we close the key circuit, which had been formerly open, then, since this is not conjugate to the battery circuit, the external resistance opposing the battery is reduced; hence its electromotive force falls, the current through the galvanometer is altered, and the deflection of the needle alters. At the same time there is a current owing to the fact that AC - BD is not exactly zero. These two effects may either conspire or oppose each other. No data, so far as we know, have been obtained which would enable us to tell how quickly this fall in the electromotive force of any given battery comes on. In practice we see a sudden jerk of the galvanometer, and then a slow swing. The former is due to the deviation of the bridge from balance, and the latter to the alteration of the electromotive force. It is easy to decide which is which, for the direction of the former can be changed by making AC - BD positive or negative, while the direction of the latter is not affected in this way. This disturbing effect is very great with one-fluid batteries; it would, for instance, be a hopeless undertaking to measure in this way the resistance of a cell of Smee while sending a large current. The effect is not so great with a Daniell’s cell, and can be reduced ad libitum, by introducing metallic resistance into the battery circuit. The effect having been thus reduced within reasonable limits, we operate thus: —Arrange the bridge until the deflection owing to. deviation from balancers opposite to that due to the change in the electromotive force; then, by gradual adjustment, work down the initial jerk to nothing, so that the needle appears to start off on its slow swing without any perceptible struggle. When this state of matters is reached, there is a balance, and B = η^-. Then subtracting fromB the resistance put into the battery circuit, we get the resistance of the battery. Of course this does not solve the problem of finding the resistance of any battery sending any current; but we believe that as much can be done in this way as in any other. Various modifications of Mance’s method have lately been proposed, but their practical advantages over the original method have scarcely as yet been established.

On Resistance in General.

We have drawn no distinction between statical and dynamical electricity in our application of Ohm’s law, and no such essential distinction has ever been proved to exist. In proportion as a body is a good conductor for galvanic electricity, it is a bad insulator for statical electricity. In general, however, bodies which are good enough insulators to retain a charge of statical electricity are so bad conductors that it is with difficulty that we can compare their conductivities by means of the voltaic current. On the other hand, it is difficult by means of statical electricity to compare satisfactorily the conductivities of very good conductors. Determinations of the last-mentioned kind have, however, been made by Riess (vide infra,—Heating Effects), and the results agree with those obtained by other methods. The insulating power of a substance depends practically to a great extent on the nature of its surface. The dissipation of statical electricity by insulating supports is due, in most cases, almost entirely to the conducting power of a thin surface layer of moisture condensed from the atmosphere, or of some product of chemical decomposition caused by exposure to the air, or of dust or other foreign matter accidentally deposited. As far as high specific resistance is concerned, paraffin, shellac, ebonite, and glass at ordinary temperatures would all be about equally good insulators; but in practice they stand in the order in which we have named them. Paraffin and shellac surpass the other two in their power of preserving for a long time a clean dry surface; ebonite is very good for a time, but ultimately its surface becomes covered with a layer of sulphuric acid, arising from the decomposition of the material; glass, again, is very hygroscopic, although white flint glass, when kept dry by artificial means, is said to be one of the best insulators known.

Highest in the order of conductivity stand the metals and their alloys. In this class of bodies the passage of the electric current is unattended by chemical decomposition, and the conductivity decreases as the temperature increases. Along with the metals may be ranked a few other bodies, which have anomalous conductivity,but are not decomposed; such as graphite, red phosphorus, chloride and oxide of lead under the melting-point,various sulphides and selenides, tellurium, and selenium. In the great majority of the bodies included in this supplementary class the conductivity increases with the temperature; the last two present several anomalies, to which we shall refer farther on.

A second class of bodies is formed by those which are decomposed by the electric current. The specific conductivity of these is much lower than that of the metals, and it increases when the temperature is raised. To this class belong, when in solution or in the melted state, most simple binary compounds composed of equal equivalents of two elements, and compounds derived from these by “double decomposition” (see, however, art. Electrolysis); also some sulphides which have an anomalous conductivity’, and glass and some bodies like it, which in the melted state, and in the soft state preceding fusion conduct as electrolytes.

Non-conductors, on the other hand, are:—All gases and vapours, whether at ordinary pressures or in what is called a vacuum, diamond, sulphur, amorphous phosphorus, amorphous selenium, fluid chlorine, bromine, solid and melted iodine, bichloride and biniodide of tin, sulphuric anhydride, solid silicic acid, oxide of iron, oxide of tin; most compounds that are not binary, that is, do not consist of an equal number of equivalents of two components, e.y., many organic compounds—etheric oils, resins, wood fibre, caoutchouc; also “binary compounds” in the solid state. To these may be added pure water, pure hydro chloric acid, <fcc., which are very bad conductors, if not absolute non-conductors.

Before leaving this part of our subject, it will be interesting to throw together a few of the general principles that have been arrived at, and to give a few numerical results, which will convey to the reader an idea of the position of the different classes of bodies in the scale of conducting power. For farther details we refer to Wiedemann’s Galvanismus.

Metals.—(1.) It was remarked by Forbes that the order of conductivity is the same for electricity as for heat. The measurements of Wiedemann and Franz have established that the ratio of the conductivities for heat and for electricity is very nearly constant, not only for pure metals, but also for alloys. (See Wiedemann’s Galvanismus, bd. i. § 194.)

(2.) The conductivity of the pure metals decreases as the temperature rises from 0° to 100° C., the rate of decrease becoming smaller towards the upper limit. Matthiessen expresses the conductivity by the formula fc=⅛ 0 (l - aθ + βθ 2 ), where ⅛ 0 denotes the conductivity at 0° C., 0 the temperature, and a and β constants. He found that a and β had nearly the same value for all pure metals in the solid state, with the exception of thallium and iron, and gives as the mean values for pure metals α=0 ’ 00376470, β = 0 - 0000083402. The values fo. iron are α = 0’0051182, 0 = 0'000012915; for mercury, a='0007443, 0 = O ’ OOOOOO8263. Although there can be no doubt about the general agreement iu the formulae for the different pure metals, yet the actual formula arrived at is purely empirical and must be used only between 0 β and 100° C. If we carried its application beyond, it would give a minimum conductivity for pure metals about 300° C. The direct experiments of Miιller and Siemens give no indication of such a minimum. The latter represents the results of his experiments (extending in some cases as far as 1000° C.) by means of the formula r = αV ’ T + 0T- γ, wherv r is the specific resistance, T the ai» solute temperature, α, β, y constants. Relying on a formula ot this kind for platinum, Siemens has .constructed a pyrometer for determining the temperature of furnaces by means of resistance measurements.

(3. ) As we have seen, the specific resistance of pure metals goes on increasing continuously as the temperature rises. At the melting-point there is a sudden rise in the resistance, and after that the resistance goes on increasing with a smaller température coefficient than before. This is in accordance with the fact, that both the specific conductivity and temperature coefficient of mercury are smaller than those of the other metals in the solid state. Bismuth and antimony are exceptions to this nιle, in that there is a sudden decrease of resistance at the melting-point. According to the results of L. de la Rive, the resistance of metals in general is about doubled in passing the melting-point. We should therefore expect the specific conductivity of frozen mercury to be about 3'31, that of silver being 100.

Alloys.—(1.) Matthiessen fouu? that the metals could be divided into two classes, according to th conducting properties of their alloys; a. Lead, Tin, Cadmium, and Zinc.

β. Most of the other metals—Bismuth, Antimony, Platinum, Palladium, Iron, Aluminium, Sodium, Gold, Copper, Silver. *

Let V, v' be the volumes, s, sl the specific gravities, k, lc , the conductivities of the two components of any alloy; and let ^s ≈ — - ⅛-, t> + v and k be called the mean specific gravity, and mean v + v conductivity of the alloy. Then alloys of any one metal of class a, with any other of the same class, have very nearly the mean specific gravity and conductivity calculated by the above formulae.

Alloys of a metal a with a metal β have specific gravity and conductivity always less than the mean. If a metal a is alloyed with a considerable percentage of β, the conductivity is not much altered, but if a metal β be alloyed with even a very small quantity of a, the conductivity is greatly reduced.

Alloys of the metals β among themselves have in general a conductivity much inferior to that of either component. The conductivity remains constant through a considerable range of percentage, but rises very quickly as the percentage of either metal approaches 100. This property is very marked in an alloy of gold and silver. Matthiessen recommended an alloy of two parts by weight of gold to one of silver for the reproduction of the standard of resistance. The resistance of such an alloy would be very slightly affected by small variations in its composition.

Mercury, and melted metals generally, are not subject to the foregoing laws. A very small percentage of another even worse conducting metal raises the conductivity of mercury, but the addition of larger quantities of the foreign metal lowers the conductivity.

(2.) The formulée for the temperature variation for alloys of the metals α among themselves agree very closely with the mean formulae calculated from the volume percentages.

If P denotes the fraction of itself by which the_conductivity at 0° exceeds that at 100° for an average pure metal (P = 0 ’ 29307), and P the same fraction, observed in the case of any alloy for which the observed and mean or calculated conductivities at 0° and 100° are k β , ⅛ loo , and k 0 , F 10β -then, according to Matthiessen, the following relation holds for alloys of metals a among themselves, and metals 3 among themselves:— _

P: P :: ⅛ 100 : 4' 100 , or, which is the same thing, R o , &<n, denoting resistances,

Rιoo ~ IL” r ioo - bo ·

For alloys of a with β, the observed value of P is in general greater than that calculated by this formulae.

' Other Physical Conditions affecting the Resistance of Solid Bodies. —Besides temperature, a variety of other circumstances affect the specific resistance of metals. Äs a general rule, metals are worse conductors in the hard than in the soft state. Tempering steel increases its resistance considerably, but subsequent heating and gradual cooling reduces the resistance again. The resistance of a wire stretched by a weight is increased more than can be accounted for by the mere decrease of the section. Winding on a bobbin has the same effect. The finer a metal is drawn into wire, the greater is its specific resistance in the case of iron, the smaller in the case of copper. Magnetization has also in certain cases been found to affect the resistance. These effects were studied by Sir William Thomson; the results of his researchas are given in his Bakerian Lecture, Phil. Trans., 1856. The experiments are very instructive, and many of them well worth repeating now that we have more delicate apparatus. The most curious case of alteration of resistance is that of tellurium and selenium. We have already mentioned that selenium in the amorphous state is a non-conductor. After continued heating it passes into the crystalline state and conducts. Sale found that the conductivity of this crystalline form of selenium is greatly affected by light, and that, too, differently by light of different colours. Prof. W. G. Adams has lately made a series of experiments on the subject, and concludes that there is an action of light, which varies as the square root of the illuminating power, and is distinct from any heating effect. He found the resistance of selenium in one case diminished by a fifth when it was exposed to the light of a certain paraffin lamp; the change in tellurium under similar circumstances was ^⅞ δ th. He found that the passage of a strong current through selenium sets up a kind of polarization, which opposes a current in the same direction as that which produced it, and aids a current in the opposite direction. This led him to suspect that the action of light might of itself start a current in tho selenium, and he found that under certain circumstances this is the case.

Fluids.—The verification by the experiments of Kohlrausch and Nippoldt of Ohm’s law for electrolytes, through a λvide range of electromotive force, has greatly increased the interest of all data relating to the resistance of this class of conductors. We have no difficulty in working with electrolytes whose composition and physical state is perfectly definite, a thing next to impossible in the case of solids. Hence the resistance of an electrolyte lias, far beyond the resistance of a solid metal, a value as datum for physical speculations concerning the ultimate properties of matter, which underlie Ohm’s law. We refer the reader to Wiedemann’s Galvanismus for an account of the earlier results in this department of Pouillet, Hankel, Becquerel, Horsford, Wiedemann, Becker, Lenz, and Saweljew. We recommend to his notice particularly the careful experiments of Beetz on zinc sulphate (his temperature determinations are the most extensive of the kind), also the researches of Paalzow, who examined the conductivity of various mixtures of two solutions, the conductivities of which had been separately determined. He finds that if R and R ’ be the resistances of the components, the resistance of the

• · RR ’ · ,. . . , mixture is not jj7jyjj5 s0 that the current is not divided between the liquids as if they were metals in multiple arc; nor is it the mean of R and R ’, but it lies nearer the smaller of the two. A similar result was arrived at by Ewing and Macgregor.

Kohlrausch and Grotrian have made the most recent as well as the most extensive investigations; and we shall best describe the present state of scientific knowledge on this subject by giving an analysis of their results and conclusions. Their experiments deal with the chlorides of the metals of the alkalies and alkaline earths. Kohlrausch has also examined a number of the commoner acids. For convenience we have transcribed the diagram given by Kohlrausch, which embodies certain of the results obtained by himself and Grotrian. Fig. 1 of the diagram gives the conductivities (⅛, g ) at

18° C.; the ordinates represent ⅛ 18 × 10 5 , except for acetic and tartaric acid, where they represent ⅛ 18 × 10 7 and ⅛ 18 × 10 6 respectively, the abscissae represent percentages by weight in the solution of HC1, H,SO 4 , NH 4 C1, &c. In fig. 2 the values of the temperature coefficient ^^^forl8°C. are given by the ordinates, .the abscissae being percentages as before. For convenience of drawing the coefficient of acetic acid is decreased by 0 - 01.

The curves which appear in the diagram include all the distinct varieties; and it will be seen that in all cases the conductivity varies continuously with the concentration, an approach to zero for infinitely weak solutions being indicated in all cases. The chlorides may be divided into two classes. (1.) CaCl. and MgCl. reach maximum conductivities 1968 × 10~ 8 , 1310 × 10“®, at 18° C. for percentages 24 and 19'8 respectively, in each case short of saturation. Li Cl probably does the same, and NaCl appears to reach a maximum between 23 - 9 p.c. and its saturation percentage 26 - 5. (2.) KC1, NH 4 C1,

SrCl i , and BaCl 2 increase in conducting power up to the point of saturation.

Taking the best conducting solutions, the order of conductivities is NH 4 C1, KC1, NaCl, LiCl, CaCl 2 , SrCl 2 , BaCl 2 , MgCl a , the alkaline chlorides heading the list. A 25 p.c. solution of NH 4 C1 is in fact half as good a conductor as the best acid solution known.

It was found that, if the conductivity for small percentages be represented by k = κp - κ'p l , so that κ may be called the specific conductivity in watery solutions, then κ varies inversely as the specific gravity, that is, directly as the “specific volume.”

The temperature coefficients for the chlorides are very nearly independent of the temperature. There is a slight increase for higher temperatures, which is most marked in the case of highly concentrated and viscous solutions of CaCl 2 , MgCl 2 .

For weak solutions the coefficients are all very nearly equal; at 18° C. the extreme value for 5 p.c. solutions lies between ⅛ (for LiCl) and ⅛ (for NH 4 C1). There is a tendency, as seen by the curves, to a value 4 ⅛, or ‘022 for very weak solutions. It will be noticed (see table below) that this coefficient is much larger than - 0039, which is about the corresponding number for a pure metal.

When the percentage is increased from five upwards, the temperature coefficient for 18° C. decreases at first for all the chlorides; it reaches a minimum for NaCl, CaCl 2 , MgCl 2 , which belong to class (1); but there is no minimum for KC1, NH 4 C1, BaCl 2 , which belong to class (2), and have no maximum conductivity.

The acids investigated were nitric, hydrochloric, sulphuric, phosphoric, oxalic, tartaric, and acetic. In every case, except that of oxalic acid, a maximum conductivity was obtained. The order in which we have named the acids is that of the conductivity of the best conducting solutions at 18° C.; for the first three we have respectively t 18 10 8 = 73 30, 7 1 74, 69 1 4, the corresponding percentages being 29 - 7, 18 - 3, 30 , 4, so that the maxima are very nearly equal, and the maximum percentages not far apart. The curve for sulphuric acid is exceedingly remarkable. Between 0 and 100 p.c. of H 2 SO 4 , it shows two maxima. The first minimum occurs at the percentage corresponding to the hydrate H 2 SO 4 + H 2 O. The conductivity corresponding to 1I 2 SO 4 is also a minimum; for when SO 3 is added, causing supersaturation, the conductivity again increases, there must therefore be at least one more maximum, since melted SO 3 is a non-conductor. There is no peculiarity in the curve corresponding to the hydrate 2H 2 O + H 2 SO 4 , which is distinguished from H 2 O + H 2 SO 4 in not being crystallizable. A striking similarity in the case of sulphuric and acetic acid is remarked between the curves of resistance and of solidification temperature; wherever the latter is high, the former is so also; there is a maximum in both casee for H 2 O + H 2 SO 4 and for II 2 SO 4 , and a minimum in both cases near 92’5 p.c.; the other minima do not agree so well.

A remarkable relation is given, which appears to connect the resistance of the monobasic acids HC1, HBr, HI, and HNO 3 . If any percentage be multiplied by the specific gravity of the solution, and divided by the molecular weight of the acid, the result is the number of molecules {ri) in unit of volume of the solution. On forming a table of resistances with n for argument, it was found that for solutions with the same n, whether of HC1, HBr, III, oι HN0 3 , the conductivity is the same. This appears very clearly from the dotted curve in fig. 3 of the diagram, calculated from the different acids, the regularity of the curve, and in parts the coincidences, are very marked. This result may be stated thus :—Zn solutions containing an equal number of molecules, whether of HN0 3 , HC1, HBr, or HI, the components of electrolysis under equal electromotive forces pass in opposite directions with equal relative velocities.

The temperature coefficients for the four monobasic acids are nearly equal, and nearly independent of the concentration. The same increase of temperature coefficient with increase of concentration as was noticed in the case of viscous chloride solutions appears also in the viscous acid solutions of phosphoric, tartaric, and sulphuric acid. It is also found that where the conductivity is a minimum, the temperature coefficient is correspondingly great; so that, with increasing temperature the maxima and minima tend to get smoothed out. It appears also that the proximity of the maxima for H a SO 4 , HNO 3 , HC1, becomes more marked as the temperature rises.

The existence of the maxima in most cases, and of the minima in the sulphuric acid curve, led Kohlrausch to suggest the principle that no stable chemical compound in a pure state is a conductor, and that mixture of at least two such compounds is necessary for conduction. He mentions many instances of this principle, e.g., water, sulphurous acid, carbonic acid, acetic acid, melted boracic acid, chromic acid, anhydrous SO 3 , &c. In a recent paper he gives some very interesting results concerning the conductivity of pure water and other bad conductors. The lowest conductivity he got for water was 71 (Hg = 10 12 ). This was after careful purification and repeated distillation in glass, and finally in platinum vessels. After standing under a'glass bell jar for 4’3, 20, 78, and 1060 hours, the water rose in conductivity from 78 to 133, 350, 850, and 3000 respectively. He calculates that, if pure water were a non-conductor, the presence of 0’1 mgr. per litre of HC1 would be sufficient to account for the observed conductivity. He also found conductivities for SnCl 4 (<) 200, alcohol (commercial distilled) 30, acetic acid (glacial melted) 4, ether (<) - 8. Among recent researches of interest may be mentioned Braun’s attempt to measure the conductivity of melted salts, and Grotrian ’ s on the relation between the viscosity and the electric conductivity of electrolytes. For the speculationsof Kohlrausch, Hankel, Beetz, Wiedemann, and Quincke on the ultimate nature of electrolytic resistance, see the papers of the first-mentioned, or Wiedemann’s Galvanismus, Bd. i. § 434 sqq.

Gases.—We are not aware that any experiments have hitherto established that any gas or vapour at ordinary temperature and pressure is a conductor. Boltzmann has arrived at the negative result that air at ordinary temperature and pressure must have a specific resistance at least 10 26 times that of copper. Sir William Thomson has, we believe, arrived at a similar result for steam; and recent experiments by Prof. Maxwell on air, steam, mercury, and sodium vapour (at high temperatures) have led him to a similar negative conclusion. It was found, however, that the heated air from a Bunsen’s burner conducts remarkably well. 7 The so-called unipolar conductivity of flames presents many anomalies, which have beeu examined by various experimenters. For the literature see F. Braun, Pogg. Ann., 1875.

It would appear, therefore, that the loss of electricity from insulated conductors at moderate potentials, observed by Coulomb and Riess, cannot be due to conduction or convection by the air, but must arise almost wholly from the insulating supports. Warburg, who has experimented much on this subject, appears to be of the same opinion {vide Boltzmann, l.c. p. 415). Varley lias lately investigated the passage of the current of a large number of Daniell’s cells through a Geissler’s (hydrogen ?) tube. He found that it required 323 cells to start the current, but that once it was started it could be maintained by 308 cells; the current which flowed was proportional to the excess of the number of cells over 304. Thus, for 317 = 304 + 13 the current was proportional to 25⅛, for 330 = 304 + 26 it was proportional to 51. Accordingly, if E o be a constant, and R another constant (the resistance of the gas?) we get for the electromotive force E, required to send a current I, E = E β + RI. E o is analogous to the electromotive force of polarization. For further details about the resistance of dielectrics we refer the reader to Maxwell’s Electricity and Magnetism, vol. i. § 366 sqq.

The following table will give an idea of the conducting power of different bodies; r denotes the specific resistance in C.G.S. units (to reduce to ohms divide by 10 9 ); a is the percentage of itself that r increases in the case of metals and decreases in the case of electrolytes per deg. C.; t is the temperature at which r is given. [table]

On the Passage of Electricity through Insulators.

Hitherto we have divided bodies into conductors, through which electricity passes under the influence of any electromotive force, however small, and non-conductors or insulators, through which electricity will not pass, no matter how great the urging force. In practice, however, when the value of the electromotive force reaches a certain limit, electricity does pass through a non-conductor. A discharge of electricity taking place suddenly in this way through a nonconductor is called a “disruptive discharge.” The power of a non-conductor to resist up to a certain limit the passage of electricity through it has been called its dielectric strength. The dielectric strength of any medium is greater the greater the electromotive force it will stand, when placed say between two parallel metal plates arranged in a given way, before it is broken through by the disruptive discharge. We shall by and by attach a definite quantitative signification to the term, but the general notion will be sufficient for the present. - .'

Although it may be found when both phenomena have been more fully analysed, that conductive and disruptive discharge are really two different aspects of one and the same phenomenon, yet for the experimenter they are two distinct things, which must not be confounded.

This would be the place to set forth the quantitative relations which regulate the electromotive force required to produce disruptive discharge, the quantity of electricity that passes under given circumstances, and the dielectric strength of different media; in fact, to lay down for disruptive discharge a law corresponding to the law of Ohm for metallic and electrolytic conduction. The present state of electrical science, however, does not permit us to do this in a satisfactory manner. Experiment has not as yet led to a single dominant principle, like Ohm’s law, which will account for all the phenomena of disruptive discharge. The best theory of the subject is Faraday’s, which will be gone into under “disruptive discharge in gases.” Observation and experiment, on the other hand, have been occupied for the most part with the various transformations of energy which accompany the disruptive discharge. We prefer, therefore, to discuss the whole matter under the single head “disruptive discharge.”

TRANSFORMATIONS OF ENERGY ACCOMPANYING THE ELECTRIC CURRENT.

Under this head we propose to discuss (to use a word of Rankine’s) the energetics of electricity. It may be objected that this heading might have been put over a good deal of what has gone before, and we shall, for convenience, treat certain matters under it which, in a strictly logical division, would have found a place elsewhere. If we had formed a definite conception of what we call electricity—had, for instance, assumed that it is a material fluid, having inertia like other fluids, then no doubt the energetics of the subject could have been much extended. As it is, we think that advantage is to be gained by associating in our minds the experimental laws which we are now to arrange under the above heading.

We shall consider (1) the heat developed in metallic and electrolytic conduction, and at the junctions in heterogeneous circuits; (2) the mechanical, sound, heat, and particularly light effects accompanying disruptive discharge; (3) the energy of magnetized iron and steel, and of electric currents in the neighbourhood of the electric current (electromagnetism); (4) the energy of the electrotonic state, or electrokinetic energy (magneto-electric induction). In this list ought to be included the potential energy of chemical separation, which would come under the head of electrolysis. At present, however, electrolysis is quite as much a chemi cal as an electrical subject, and it has been found convenient to treat it in a separate article (see Electrolysis). Some points in connection with it have already been touched upon, and a few more will come up in (5), which treats of sources of electromotive force, and deals with the question, whence comes the energy which is evolved in the voltaic circuit? a question the answer to which is for the most part experimental and practical—the only one, in fact, that the state of electrical science permits us to give.

Heating Effects.

It is easy to show, by a variety of simple experiments, that a current of electricity heats a conductor through which it passes. In the case of moderately strong currents the heat developed is perceptible to the touch; the wire may, in the case of very strong currents, be raised to a white heat; it may melt, and even be volatilized. In the case of very weak currents, the heating effect may be demonstrated by passing the current through the spiral of a delicate Breguet’s thermometer. We find, when we examine the experimental data on the subject, that heating effects may be conveniently divided into two distinct classes. In the first of these the fundamental law is that the development of heat in any part of a linear circuit varies as the resistance of that part multiplied by the square of the current. In the second class the development of heat varies as the first power of the current. The heating effects of the first class are obviously independent of the direction of the current, and are irreversible; and the more we examine them the more they appear to correspond to the loss of energy by the frictional generation of heat in ordinary machines. In the language of the dynamical theory of heat, the part of the energy of the electric current which disappears in this way is said to be dissipated. The effects of the second class change their sign when the direction of the current is changed; so that, if anywhere there was evolution of heat when the current flows in one direction, then, when the current is reversed, there will be absorption of heat to an equal extent. We shall find that we have great reason to believe that such effects are strictly reversible. In order to get a satisfactory foundation for the simple theoretical views which we have thus indicated, it is essential to be able to separate the two classes of effects. Now, this is possible , to a very great extent even in practice. The effects of the first class increase much more rapidly with the strength of the current than those of the second, so that, by sufficiently increasing the current, we can make the effects of the second class as small a fraction of the whole heating effect as we please; while, on the other hand, by sufficiently decreasing the current, the preponderance of the second class may be increased to any desired extent. We shall in what follows suppose the two classes of effects separated in this way.

Discharge of Statical Electricity.—One of the earliest attempts to study the heating effects of the electric discharge was made by Kinnersley. He constructed an thermoelectrometer, which consisted of a closed glass vessel, in which were fixed two metal balls communicating with electrodes outside the vessel. The bottom of the vessel was filled with a little coloured fluid, which communicated with a tube having a vertical arm rising outside the vessel. When a spark passed between the balls, the heat developed caused the air to expand and force the liquid into the vertical tube, the rise of level in which indicated the degree of expansion, and, by inference, the amount of heat developed in the spark.

Sir Wm. Snow Harris revived this instrument of Kinnersley’s, and improved it by stretching a fine wire between the terminals inside the vessel, so that the heat measured was now that evolved in a metallic conductor.

With this improved instrument he made a number of valuable experiments on the heating of wires by the discharge of a Leyden battery, whose charge was measured by a Lane’s electrometer. Assuming that the heat developed varies inversely as the conductivity of the wire (which is not the case), he arranged the metals in a series which agrees with that given later by Riess, although the numbers given do not properly represent the conductivities owing to the erroneous assumption on which they are deduced. Harris observed that the specific conductivity of alloys is often less than that of either metal, and that a very small admixture of another metal considerably reduces the conductivity of pure copper. He also arrived at the result that the amount of heat developed in a wire varies as the quantity of electricity which passes in the discharge, but seems to have concluded that the amount of battery surface used had no effect.

Riess made two very important improvements on the thermoelectrometer by substituting spirals for the straight wire of Harris, and by inclining the tube containing the liquid so as to be nearly horizontal. The sensibility of the instrument was thus greatly increased. Riess took up the whole question of the heating of wires, and investigated it thoroughly.

The actual instrument which he used is represented in figure 27 (taken from bis Reibungselcctricit<itj It consists of a glass tube of narrow bore, 16 to 17 inches long, to which is blown a glass globe 3 to 4 inches in diameter. This tube is partially filled with some coloured fluid which confines the air in the globe , a wide reservoir at the other end of the tube allows the fluid to accumulate without sensible change of level. The stand of the instrument consists of two pieces hinged together, so that the tube can be placed at a small inclination to the horizon. The rest of the instrument will be understood from the figure. Details concerning the manipulation will be found in the Reibungseleclricilät, Bd. i. § 410. When the fine wire is heated by a current of electricity, the heat developed is divided between the wire and the air; the expansion very quickly reaches a maximum, and the level of the liquid in the fine tube becomes stationary for a moment. If m be the number of scale divisions between its original and final positions, we have (see Riess, l.c., or Mascart, t. i. § 325)

T-wιA^l and H=rnA(CW + B) .... (1), where T denotes the amount by which the temperature of the wire would have risen had no heat left it, and H the whole amount of heat developed by the current. C and W are the mean specific heat and weight of the wire, and A and B constants, which depend on the make of the instrument, and on the initial temperature and pressure of the air.

A very convenient form of thermoelectrometer, called the thermomètre iuscripteur, has been used by Mascart (Z.c.).

The alterations of pressure are registered automatically on a revolving drum, after the manner of the pulse-registering instrument of Marey. One advantage of this instrument is that it gives a representation of the course of the temperature in the apparatus.

In most of his experiments Riess used batteries of Leyden jars. The jars were all as nearly as possible alike, and the inner armatures were in general connected together. The quantity of electricity given to the battery was measured by means of a Lane’s jar, the balls of which were placed at a distance of about a line apart. The battery was then discharged through the thermo-electrometer along with any external circuit connected with it.

It is of great importance in such experiments as we are now describing to examine what happens at the place where the circuit is closed. This closure is effected by bringing two metallic balls into contact. But before contact is reached, a spark passes in which sound, light, and heat are given forth,—in a word, energy evolved. When the resistance of the circuit is small, this spark passes at a considerable distance, and is very intense, no matter how quickly the conductors are brought together. The energy consumed in this case is a considerable fraction of the whole energy given out by the discharge. If, however, the resistance of the circuit through which the discharge takes place be considerable, the electricity takes longer to accumulate sufficiently to raise the electromotive force between the balls to the discharging limit. We may, therefore, by operating quickly, get the balls very nearly in contact before the spark passes. In this case the spark is much less intense, and the fraction of the whole energy which appears in it is very small. Riess made some very valuable experiments on this point. He arranged an air-break in the circuit of the thermoelectrometer, which he could widen or narrow at pleasure, and discharged his batteries through this circuit in the usual way. He found that as the gap is widened the amount of heating in the thermometer is at first increased, but after a certain length of break is attained it decreases again. It must be remembered that we have now two air breaks in our circuit of discharges, the discharging break and the inserted break. One effect of the inserted break is to diminish the intensity of the spark at the discharging break, and cause a decrease of the energy which appears there. On the other hand it makes the discharge of the battery incomplete, so that part of the potential energy is not exhausted. It is very likely to the opposition of these two effects that·the peculiarity observed by Riess is due. Mascart has observed a similar phenomenon in disruptive discharge through oil of turpentine. At all events Riess showed that, when the inserted break was not longer than ∙⅛ths of a line, the heating in the thermometer was the same as when there was no break at all. Hence, if we make the resistance of our circuit so great that the spark at the discharger is not longer than τ ¾ths of a line, the energy consumed there may be neglected.

The resistance of the connections belonging to the battery and the thermometer were always very small compared with that of the thermometer wire, and the wire, if any, inserted outside the thermometer; so that, if the resistances of these be R and S, the resistance of the whole circuit may be taken to be R + S. The law to which the < experiments of Riess led can be expressed by means of the formula h =ETs q < 2 >∙ where Q is the amount of electric potential energy which has disappeared, and H the amount of heat (measured by its dynamical equivalent) developed in the wire of the thermometer, whose resistance is S.

In the case of the complete discharge of a battery of n jars, each of capacity C, if q be the whole charge, we get immediately, from (48) of Mathematical Theory (p. 34), q"

^ = 2Cn’ and π 8 ¾ ,8 . (3)

h- R + S' 2Cn 1 1

Hence, if we keep the thermometer and inserted wires the same, the thermometer indications will be proportional to a 2

— f or, in words—the heat evolved in the whole or in any r nJ

given part of the circuit is proportional to the square of the battery charge directly, and to the number of jars (i.e., to the battery surface) inversely.

If the thermometer wire remain the same, while the length, section, and material of the inserted wire is varied, then, r being the specific resistance, l the length, and p A)'l the diameter of that wire, R = — 2 . Then, according to

(3), the heat developed in the thermometer is given by

H-^- τ (4),

1 + B⅞ 0

where A and B are constants.

If, again, we use two wires of the same material of lengths l and l' and diameters p and p', and make two observations with these for inserted and thermometer wires respectively and vice versa, then, if H 1 and H 2 be the heat evolved in the two cases,

H 1 = l , P t , (5) h 2 Z7≡ (5) ’

since R + S is the same in the two cases.

When the discharge is not complete, we have only to substitute for Q in (3) the appropriate expression for the exhaustion of the electric potential energy. Similarly we may find the heating effect caused by the discharge of a battery of jars arranged in series and charged by cascade in Franklin’s manner (p. 35). If we discharge through a multiple arc, we may assume that the discharge divides itself between the branches in the ratio of the conductivities, so that the conductivity of the whole arc is the sum of the conductivities of its parallel branches. On these principles it is easy to calculate the heat generated in the whole circuit or in any branch of the arc.

All the cases we have alluded to were treated experimentally by Riess, and satisfactory agreement with formula (2) established in every case.

By means of formula (4) he compared the specific conductivities of a variety of metals. A and B were determined, and a standard wire of platinum of given length kept in the thermometer; the wires to be compared with it were inserted in the outside circuit, and the heating in the thermometer observed. From the result the specific conductivity (in terms of platinum) of the wires could be calculated, their dimensions being known. The results agree very well with those got by other means.

Heating by Constant Current. —The heating effect of the current furnished by a voltaic battery was recognized as a distinct and often very remarkable phenomenon for a considerable time before any definite quantitative law was established regarding it. Davy experimented on wires of the same dimensions but of different materials, and found that the metals could be arranged in the following order :— silver, copper, lead, gold, zinc, tin, platinum, palladium, iron,—those standing nearer the beginning of the list being less heated by a given current than those nearer the end.

Joule was the first, however, to establish a definite law . connecting the amount of heat evolved per second with the - current strength and the resistance of the wire. He wound the wire in which the heat generated was to be measured round a glass tube which was immersed in a calorimeter. The resistance of the water is so great that we may assume without sensible error that the whole of the current passes through the wire. The temperature of the water was determined by means of a mercury thermometer immersed in the calorimeter. The amount of heat developed in the wire per second could then be found by the usual calorimetric methods. The strength of the current was measured by means of a galvanometer inserted in the battery circuit along with the wire. By experiments of this kind Joule established that the amount of heat generated in a given time varies directly as the product of the resistance oj the wire into the square of the strength of the current. So that, if we choose our units properly, we may write

II = RI 2 Z (6), where R is the resistance of the wire, I the strength of the current, and H the quantity of heat generated in time t.

The experiments of Joule were repeated with increased precautions against error by Becquerel, Lenz, and Botto. Becquerel allowed the wire to disengage heat till the calorimeter reached such a temperature that the loss of heat by radiation and convection, &c., was just equal to the gain from the wire, so that the temperature became stationary. The current was then stopped, and the loss of heat per second found by observing the fall of temperature in the calorimeter. Botto used an ice calorimeter. Lenz made a series of very careful experiments with a calorimeter, in which the liquid used was alcohol, which is a much worse conductor than water. He first cooled his apparatus a few degrees below the temperature of the surrounding air, and then allowed the current to generate heat in the wire till the temperature of the whole calorimeter (which was kept uniform by agitation) had risen to an equal number of degrees above the temperature of the air. The current was then stopped, and the time t which it had flowed noted. According to Joule’s law, iRI 2 ought to be constant, and it was found to be so very nearly. A very convenient instrument for demonstrating and measuring the heat generated by the electric current in a wire is the galavano-thermometer of Poggendorif, which consists simply of an alcohol thermometer with a large bulb, into which is let a spiral of fine wire. The heat generated is deduced from the expansion of the alcohol, which is measured by means of a scale fastened to the stem of the thermometer. The value of the graduations is found by comparison with an ordinary thermometer. The thermoelectrometer of Riess might also be used in a similar way.

Heating in Electrolytes.—Joule’s law applies also to electrolytes. The phenomenon, however, is not so simple as it generally is in the case of metallic conductors. Disturbances arise, owing to the heat evolved aud absorbed in the secondary actions that take place at the electrode; and superadded to this we have in all probability an absorption or evolution of heat corresponding to the Peltier effect between different metals, of which we shall have to speak directly. Joule eliminated these disturbing influences by using a solution of copper sulphate with copper electrodes. In this case copper is dissolved from one electrode and deposited on the other, so that if we except the slight difference in the states of aggregation of the dissolved and deposited copper, the secondary processes are exactly equivalent, and must compensate each other. Joule found that in a certain solution of CuSO 4 5’50 units of heat were generated in a certain time, λvhile in a wire of equal resistance 5 - 88 units were generated by an equal current in the same time. In a similar manner E. Becquerel found that a current, which would produce a cubic centimeter per minute of explosive gas, generated in certain solutions of CuSO 4 and ZnSO 4 0 - 213 and 0 ’ 365 units of heat; while the same current would have generated in wires of equal resistance 0 ’ 26 and 0 - 32 units respectively.

Reversible Heating Effects.—Peltier was the first to discover an effect of this nature. He found that, when an electric current passes over a junction of antimony with bismuth, the order of the metals being that in which we have named them, there is an evolution of heat at the junction; and, when the current passes in the opposite direction, there is an absorption of heat, so that the temperature of the junction falls. Here, therefore, there is an effect which cannot vary as the square of the current strength, but must be some function of the current strength, whose principal term at least is some odd power.

The Peltier effect, as it is now called after its discoverer, may be demonstrated by inserting a soldered junction of antimony and bismuth into a Riess's thermoelectrometer. When the current goes BiSb, the fluid will rise in the stem, indicating absorption of > > heat; when it goes Sb Bi, the fluid will fall, indicating evolution of heat. Or we may use Peltier’s cross, which consists ot two pieces, one of bismuth BB ’, and the other of antimony AA ’, soldered together in the form of a cross (fig. 28). A and B are connected by a wire through a galvanometer G. A ’ and B ’ are connected with a battery C through a commutator D, by means of which the current can be sent either from A ’ to B ’ or from B , to A ’ through the junction. The thermoelectric current indicated by the galvanometer shows that the junction is heated iu the first instance and cooled in the second.

By leading the current of a Grove’s cell for five minutes through a BiSb junction, Lenz succeeded in freezing a small quantity of water which had been placed in a hole in the junction, and previously reduced to 0° C. The temperature of the ice formed fell to - 4∙5° C.

The Peltier effect is different for different pairs of metals. Peltier and Becquerel found that the metals could be arranged in the following order :—

» ¼

Bi, Gs, Pt, Pb, Sn, Cu, Au, Zu, Fe, Sb.

If the current pass across a junction of any two of these metals, cold or heat is generated according as the current passes the metals in the direction of the arrow or in the opposite direction; and the Peltier effect between the metals is greater the farther apart they are in the series. We shall see later on that this is none other than the thermoelectric series.

Von Quintus Icilius showed that the Peltier effect is directly proportional to ths strength of the current. He passed a voltaic current through a tangent galvanometer (serving to measure it) and a thermopile of 32 BiSb couples. The current was allowed to pass for a fixed time, then the battery was removed and the thermoelectric current of the pile measured by means of a delicate mirror galvanometer. The current of the battery heats the pile in part uniformly according to Joule’s law: this causes no unequal heating of the junction, and therefore no thermoelectric current; and in part unequally, so that one set of junctions are cooler and the other warmer than the mass of the metal: this causes a thermoelectric current, which, since the temperature differences are small (see below, p. 97), may be taken to be proportional to the temperature difference, that is, to the double of the Peltier effect at each set of junctions.

It is interesting to note the analogy here with the polarization of an electrolytic cell. We turn a battery on to the thermopile, and polarize it, as it were. Then, when we remove the battery and close the pile, we get a return current, which might be called the polarization current of the thermopile.

In general the Peltier effect is, as we have seen, mixed up with Joule’s effect, and makes itself felt by producing a disturbance at the junction. Thus Children found that, when a strong current passed through two mercury cups joined by a thin platinum wire, so that the wire became red hot, the temperature of the mercury in the cups next the + pole of the battery rose to 121° F., while in the cup next the - pole the temperature was only 112° F. Frankenheim studied the two effects together. He made a Peltier’s cross of the pair of metals to be examined, passed a current I through the cross first in one direction and then in the other, and determined by means of a delicate galvanometer the thermoelectric current generated in each case, which is very nearly proportional to the heat produced. If α and b be the heat from Joule and Peltier effects respectively, and i and i' the observed thermoelectric currents, then ι=C(α + δ), i' = C(α-Z>)j whence α = (i+√)÷2C, and δ=(t-i ’)÷2C. In this way he found that a was proportional to I 2 , and b to I. Thus the whole heat developed may be expressed by αl 2 ±δl. We get in this way a verification of the results both of Joule and of Von Quintus Icilius.

Further experiments have been made on this subject by Edlund and Le Roux; and Sir W. Thomson was led by a remarkable train of reasoning to discover another reversible heating effect. We prefer to leave these matters for the present, to return to them when we consider thermoelectric sources of electromotive force.

The Peltier effect between metals and liquids and other reversible effects will also come uυ again under the Origin of Electromotive Force.

Theoretical Deduction of the Formulée.—The above for-inulae for the heat developed in wires by statical and dynamical electricity may be deduced from a common formula, which can be deduced from Ohm’s law.

Let P, Q be two points of a linear circuit, and let E be the difference between the potentials at P and Q, then, if there be no other electromotive force in the portion PQ, the work done by a unit of + electricity in passing from P to Q is E. Hence, if I be the strength of the current, so that lift units of electricity pass from P to Q iu time dt, then the amount dw of work done by the current in time dt is Elcft. But, by Ohm’s law, E = RI, hence dw=Rl 2 dt ....... (7).

Since the whole of this work is spent in heat, we may for w write H, which denotes the heat generated in PQ. If the current be constant, we get immediately H=RI⅜, which is Joule’s law (6). If the current be variable, H=∕RI 2 (ft, from which we may very easily deduce the formula for the discharge of a battery of Leyden jars. For, applying Ohm’s law to the whole circuit whose resistance is R + S, we have, if U denote the potential of the inside coatings at time t, I=— T -. . Also the capacity of each of the n jars being C, we have for the charge ∕ = nCU, and dt =nC-^ . Hence dt

H = R ∕*l 2 i ft = -⅛≤⅞ ∕U^<ft=^2ξ Σ’ J I∖ + S β ∕ dt h + S 2

_ R 7 ’, 8)

R + S ’ 2uC ' a where q and V have the same meanings as in (3). (8) agrees with

(3), except that we have reckoned the heat developed in a portion of the circuit whose resistance is R instead of S, as in (3). It appears, therefore, that the theoretical formula (7), when properly interpreted, covers both cases.

I f there were a junction of heterogeneous metals in the part PQ of the circuit, at which the potential suddenly fell by an amount ∏, then work equal to ∏ltft would be done by the current in passing over the junction, and we should have to write dW=RI 8 <ft + ∏l(ft ...... (9).

Had there been a rise of potential at the junction, we should have written - ∏ instead of + ∏. If all the work done at the junction is transfo'. , med into heat, W = H as before, and for a constant current,

H = RI 2 i + ∏R . ... (10).

The first term is Joule's, the second Peltier’s effect. Here the coefficient of the Peltier effect appears as an electromotive force. We shall return to this again.

Glowing, Melting, Volatilization, Ac.—If a wire lost none of the heat generated in it, then, for the same current, the rise in its temperature during a given time would vary as its specific resistance directly, and as the product of its specific heat and density into the fourth power of its diameter inversely. Thus, T, r, c, p, d denoting these quantities in the order named above, τ oc ⅞⅛ ·

If we have a given battery of electromotive force E, and a circuit connected with it of resistance R, and we insert a wire of length l specified in other respects as above, the current will be > where · If the s diameter of the wire be given, then S oc l · } and T <x , which is a maximum when R = S, that is, when the length of the wire is such that its resistance is equal to that of the rest of the circuit.

Owing to our ignorance of the exact law of cooling, and of the manner in which the resistance and specific heat of most metals change at very high temperatures, it is very difficult to predict beforehand to what temperature a given current will raise a given wire. It is, as may be supposed, still more difficult to predict the effect of a given discharge from a Leyden battery. According to Riess, the phenomenon of glow in this case is complicated by concomitant effects of specific nature.

If we assume Newton’s law of cooling, i.e., that the heat given out is proportional to the surface of the wire and to the elevation T of its temperature over that of the surrounding medium, then, I denoting the strength of the constant current which heats the wire, we have, when a constant temperature has been attained, ] 2 =≈ const. × Td 3 , for wires of same length and material but different diameters. If we compare the apparent brightness of the wires, by causing them to illuminate a screen at a constant distance off, and assume that the light given out is proportional to Trf, then, if two wires of diameters <∕ 1 and rf 2 have the same apparent brightness, T 1 d 1 = T 2 ⅛ and I,÷cf, =I 2 ÷d 2 . In other words, the strength of current requisite to bring a wire of given length and material to a given brightness of glow varies directly as its diameter. A law of this nature is, of course, merely a rough approximation; Müller and Zöllner, however, have made experiments which agree with it within certain limits. The method of Zöllner is interesting (see Wiedemann's Galvanismus).

The temperature of a glowing wire is very sensitive to external circumstances, such as air currents, &c. These effects may be very strikingly shown by balancing the wire in a Wheatstone’s bridge against a resistance of thick wire, a strong current being sent through the bridge.

The behaviour of the wire in different gases is very remarkable. If a wire which is glowing in air be suddenly immersed in a jar of hydrogen or coal gas, the brightness will be very much reduced, in fact, in most cases the glow will entirely disappear. This is owing to the greater cooling power of hydrogen, of which evidence is furnished by the experiments of Dulong and Petit. The cooling power of different gases was shown by Grove. He arranged a platinum wire in a glass tube, which could be filled with different gases. The current of the same battery was sent through the wire and through a voltameter. When the tube was filled with hydrogen or olefiant gas, the amount of gas evolved in the voltameter per minute was 7'7 and 7’0 cubic inches respectively. The numbers for the other gases experimented on varied from 6 - 6 to 6T. They stood in the following order:—CO, CO 2 , 0, air (2 atmos.), N, air (1 atmos.), air (rarefied), Cl. Experiments of a similar nature were made on liquids. Clausius carried out a calculation of the cooling effect of different gases, and found that the experimental results could be satisfactorily accounted for.

When the strength of the current is sufficiently increased, the wire ultimately fuses, or even volatilizes. The pheno menon is in general complicated. In air, for instance, the wire burns, and the oxidization once started may take a greater share in raising the temperature than the current does, so that the destruction of the wire may take place under certain circumstances with a current, which, under other conditions, would scarcely make it glow. When discharges from a Leyden battery are used it is very difficult, if not altogether impossible, to get melting unaccompanied with mechanical disaggregation of the wire. The reader who wishes for further information concerning these matters, will find the sources sufficiently indicated in Wiedemann, Riess, and Mascart.

This department of electricity is very fruitful in popular lecture-room experiments. We shall quote one or two of these, and refer the reader to popular treatises for more of the same kind.

On a sheet of thin card-board is pricked a design, generally’ what is understood to be a portrait of Franklin, two pieces of tinfoil are pasted on the ends of the card by way of electrodes, and between these a piece of gold leaf is laid. On the other side of the card is placed a piece of white paper or silk. The whole is then tightly screwed up between two boards. When an electric discharge is sent through the gold leaf it volatilizes, sending the disintegrated particles through the holes in the card-board. In this way an impression of the portrait is obtained.

If a current be caused to heat a pretty long thin platinum wire to dull redness, and a portion of the wire be cooled by applying a piece of ice to it, the remainder of the wire will glow much more brightly than before; whereas, if a portion be heated by a spirit-lamp, the reverse effect takes place. The reason is that the current is strengthened in the one case by the decrease of the resistance in the cooled part, and weakened in the other by the increase of resistance where the wire is heated.

When two curved metal surfaces rest upon each other, a current passing from the one to the other encounters considerable resistance at the small area of contact. The heat developed in consequence of this causes the parts in the neighbourhood to expand very quickly when the contact is made. This very often gives rise to rapid vibratory movements in the conductors. The Trevelyan rocker can be worked in this way (see art. Heat), bells rung, &c. The best known experiment of the kind is Gore’s railway. This consists of two concentric copper hoops, whose edges are worked very truly into the same plane. A light copper ball is placed on the rails thus formed, a current from two or three Groves is sent from one hoop to the other, and the ball set in motion. If the ball be very true, and the railway be well levelled, the energy supplied by the swelling at the continually changing point of contact is sufficient to keep up the motion, and the ball runs round and round, emitting a crackling sound as it goes.

The Voltaic Arc.—When two electrodes of volatile or readily disintegrate material forming the poles of a powerful battery (say 30 or 40 Grove’s cells) are brought into contact and then separated, the current continues to pass across the interval, provided it is not too great. The conducting medium appears to be a continuous supply of heated matter, suspended in glowing gas or vapour. This phenomenon seems to be more akin to the subject we are now discussing than to the disruptive discharge of which we shall speak by-and-by. The light thus generated with a large battery, especially when electrodes of graphitic carbon are used, is brilliant in the extreme. It was thus that Davy first obtained the phenomenon. With a battery of 2000 cells he obtained a luminous arc 4 inches in length, and when the carbons were placed in an exhausted receiver the arc could be lengthened to 7 inches.

The fact that the electrodes must be brought in contact in order to start the light is quite in accordance with what we know of the extremely small striking distance of even very powerful batteries. When the contact is made, the place where the electrodes touch, owing to its small section, is intensely heated; the matter begins to volatilize, and then the current is kept up by the quickly increasing cloud of metallic vapour and disintegrated matter. With a battery of 60 Grove’s cells the arc once started has a certain persistence, for we may break the current for 5 1 σ th of a second or so, and the light will start again when the current is turned on afresh. We may start the light without bringing the electrodes into contact by causing a spark from a Leyden jar or an inductorium to pass across the interval.

If an image of the voltaic arc be thrown on a screen by f means of a lens, its constitution can be examined very readily. Four distinct parts at once strike us;—first, the dazzling white positive carbon, which assumes a crater-like shape after the current has passed for some time; second, the more pointed and if-anything less brilliantly-white negative carbon, which is in general strewed with little beads of melted or at least softened carbon; third, the central core or streak of glowing matter, which has a white appearance, though it is considerably less brilliant than the carbon; fourth, the globe-shaped aureole which surrounds the whole, whose brilliancy is greatly inferior to that of the other parts, and whose colour depends on the surrounding gas. If the electrodes be horizontal, the arc is in general curved upward by ascending air currents, its form is also affected in general by the earth’s magnetic action.

The hollowing out of the positive electrode is obviously due to a transfer of matter in the direction of the current. It is very easy to prove by a variety of conclusive experiments that there is such a transfer, mainly in the direction of the current, but also in part in the opposite direction. If we take a platinum point for positive, and a platinum plate for negative electrode, the matter carried to the plate forms a series of rings on it like the colour rings of Nobili. If, on the other hand, the platinum plate forms the positive electrode, a series of slight excavations are formed where the matter has been torn away. There can be no doubt that the disintegration of the electrodes plays a very important part in the formation of the arc, for if we saturate the carbons with volatile matters, the brilliancy of the arc, the ease with which it forms, and its maximum length for given battery power are greatly increased. It is probably owing, in part at least, to the tearing away of matter at the positive electrode that the temperature there is in general highest. This effect is very marked in some cases. If we take a platinum point and a plate of the same metal for electrodes, the point glows through a considerable length when it is positive, but only at the end when it is negative. Again, when the light is generated between two platinum wires held crosswise at a small distance apart, the glowing portion is much longer on the positive electrode than on the negative.

The electric light is the only artificial light whose brilliancy can compare with the sun. Measured by its actinic properties simply, it is not so very far behind the great luminary; its spectrum is longer towards the violet, and it has accordingly great advantages when it is required to produce fluorescence (see art. Light). Its great chemical power is also shown by the readiness with which it induces the combination of hydrogen and chlorine; by means of it underground buildings, such as the catacombs, have been successfully photographed. Its illuminating powers have for a considerable time been employed in lighthouses, the current for its maintenance being furnished by powerful electromagnetic machinery, and it is now proposed to employ the Gramme machine and the electric lamp to light streets and public buildings, manufactories, <fcc. It was used for war purposes during the last siege of Paris, and in the Russo-Turkish war on the Danube; and further applications to torpedo warfare have been contemplated.

In most of these applications of the electric light it is important that the arc should be of constant length, and maintain a fixed position. Owing to the unequal consumption of the carbons, special appliances are required to secure the fulfilment of these conditions. The best known and perhaps the most efficient of the older electric lamps is that devised by Foucault. It consists of a piece of clockwork, which moves the carbons towards each other with relative speed nearly equal to that at which they are consumed. The machinery is controlled by a detent worked by an electromagnet, which is excited by the current which feeds the arc. When the carbons are too far apart the electromagnet is weakened and releases the detent. The machinery then moves the carbons until the current is strong enough to enable the electromagnet to apply the detent again. This apparatus works well enough for lecture-room and other purposes, but has not given perfect satisfaction in industrial applications. Accordingly many devices have been proposed, more especially of late years, to supersede it. One of the simplest and it would appear most effective of these is the electric candle of Jablochkoff. This consists simply of two carbons, separated from each other by a plate of kaolin. The arc passes between the carbons, and plays over the kaolin, which gradually melts away like the wick of a candle, and by its incandescence greatly helps the brightness of the light.

The heating powers of the electric arc are no less remarkable; platinum and iridium melt in it like lead, and volatilize. In this way the spectra of the glowing vapour of these metals can be projected on a screen. Almost nothing seems to resist the elevated temperature of the arc. Despretz generated it in vacuo by means of 500 to 600 cells of Bunsen, and observed pieces of carbon volatilize like a piece of heated iodine, while the carbon vapour condensed on the walls of the receiver in the form of a crystalline powder. Flint melted to a glassy mass, and boron behaved similarly, while cylinders of retort carbon softened and bent into an S-form.

The voltaic arc behaves in many respects like an ordinary electric current. It is affected by the magnet, for instance, as an ordinary current would be. Owing, however, to the variety of transformations of energy taking place, it is difficult to estimate accurately the resistance and electromotive force of the arc. Edlund made experiments which seemed to show that a certain minimum electromotive force in every case was necessary for the maintenance of a continuous arc, yet the arc does not appear to consist of a series of disruptive discharges, for its image in a rotating mirror is a uniform band. It would seem, therefore, as if polarization in some form or other were present. Edlund, in fact, found that when a galvanometer was substituted for the battery by which the arc was formed a considerable current was obtained, which might have an origin similar to that of electrolytic polarization, or be a thermoelectric effect. For further details on these and other matters connected with the electric light, we refer the reader to the admirable account of Wiedemann, Galvanismus, Bd. i. §701 sqq.), from which most of the above is taken.

Disruptive Discharge, Light Effects, <L∙c.

A definite meaning has already been attached to the term disruptive discharge; the object of the present section is to consider this phenomenon a little more closely in several particular cases. The disruptive discharge proper is in general accompanied by sound, heat, light, and mechanical effects, very often by all four. The attendant luminous phenomena have absorbed by far the greatest share of the attention of experimenters, partly, no doubt, on account of their great variety and wonderful beauty. It would be a hopeless task to endeavour, within the limits set us here, to give even a meagre summary, not to speak of a critical account, of all the experiments and observations that have been made on this subject. The scientific investigator will find sufficient guide for his reading in the three standard treatises of Riess, Wiedemann, and Mascart. Riess is particularly interesting in his account of the older experiments; Wiedemann, on the other hand, gives elaborate accounts of the more modern results of De la Rive, Plücker, Hittorf, and others.

When induction is exerted across a dielectric, we may consider the action at any point of it in one or other of two ways. We may regard the resultant electromotive force arising from the action at a distance of all the free electricity in the field as tending to separate the two electricities in the molecules of the dielectric. In this view, we might measure the dielectric strength of the medium by the value of the electromotive force, when the electricity is on the point of passing from one molecule to the next. We might, on the other hand, consider, with Faraday and Maxwell, that the dielectric is the seat of a peculiar kind of stress, consisting of a tension p along the lines of force, and an equal pressure perpendicular to them, p being equal to J^R 2 (Maxwell, vol. i. § 104). We shall adopt the latter alternative, and when we speak of tension henceforward it means — R 2 . In this view the dielectric strength 077

may be defined as that tension under which the dielectric just begins to give way. The reader who prefers the other way of looking at the matter will find no difficulty in translating any statement from the one language into the other.

We have started by considering any point of the dielectric, and it is obvious that the dielectric (supposed homogeneous) will first give way at that point which first reaches the limiting tension ·=>·; just as an elastic solid begins to give way where the stress first reaches the breaking limit. It may be proved, however, that R 2 cannot have a maximum value at any point where there is no free electricity, which shows us at once that the point at which the limiting tension is first reached must always be on some electrified surface, in general therefore on the surface of one of the conductors of the system. Disruptive discharge, thus begun at the surface of a conductor, spreads out into the dielectric. Its farther course is influenced by a variety of circumstances very hard to define in the great majority of cases.

An attempt will be made by-and-by to give an idea of the varieties of luminous discharge that arise in this way; meantime we concentrate our attention on a feature common to all disruptive discharges, viz., the definite limiting tension at which under given circumstances they begin.

Dielectric Strength of Gases.—The earlier measurements bearing on this subject were conducted under circumstances which render a compa r ison of the results with the theory, as at present developed, very difficult. Harris found that the striking distance between two balls connected with the armatures of a condenser was directly proportional to the charge of the condenser as measured by a Lane’s jar. Riess used a Leyden battery, and varied the number of jars and the charge of the battery. The balls of his spark micrometer were of diameters 5 ’ 7 and 4 ’ 4 lines respectively, while the distance between them varied from 0 ’ 5 to 2 - 5 lines. Under these circumstances, he found the striking distance to be proportional to the charge of the battery directly, and to the number of jars inversely. The results of Harris and Riess might be summed up in the statement that the striking distance between two balls connected with the armatures of a condenser varies as the electromotive force or difference of potential between the armatures. This result is purely empirical, and must not be extended beyond the experimental limits within which it was found. Even Riess’s experiments themselves show that the striking distance increases more rapidly than the difference of the potentials.

The experiments of Knochenhauer led to a similar result. Gaugain made experiments of the same kind through a wider range of striking distances, and found, in conformity with the result of Riess, that, with balls of 10 or 15 mm. diameter, the striking distance is proportional to the potential difference between the balls, when the distance between them lies between 2 and 5 millimetres. Beyond these limits the ratio of potential difference to striking distance falls off; whereas, for smaller ^distances, ft increases very rapidly. He also found that the deviation from the law of Harris and Riess is more marked when unequal spheres (3 mm. and 10 mm.) are used, and still more when a ball (3 mm. diam. used as + electrode) and a disc (35 mm. diam.) were used as electrodes. Experiments leading to similar conclusions are cited by Mascart, who finds that, for spheres of diameter 3 to 5 centimetres, the striking distance for given potential difference is sensibly the same; whereas for plates, both the striking distance and the law of the whole phenomenon is different. The same experimenter examined the striking distances between two equal balls (3 cm. diam.) from 1 mm. up to 150 mm. Taking the potential difference for one millimetre as unity, he found for 10, 20, 40, 80, 150 mm. the potential differences 8 , 3, 11'8, 15 - 9, 20'5, 23 - 3. The deviation from proportionality is obvious; the potential differences in fact tend to become constant. λViedemann and Riihlmann, in their experiments on the passage of electricity through gases (see below, p. 61), made some experiments on the influence of the form and distance of the electrodes. They used two brass balls of 13 ’8 and 2’65 mm. diameter respectively, and sent between them the discharges of a Holtz machine. The distance (δ) between the nearest points varied from 3 to 22 '3 mm. They found that the quantity of electricity (y) required to produce discharge, could be represented by the formulae y = A,-? and y = C + Dδ 2 , according as the larger o sphere formed the positive or negative electrode. The constants A, B, C, D depend on the pressure, which varied in these experiments between 25 and 60 mm. of mercury.

In most of the experiments that have just been described the effect of the form of b the electrodes and the surrounding conductors could not be estimated theoretically. Experiments in which the theoretical conditions are simple have been made by Sir Wm. Thomson. The spark was taken between two parallel plates of considerable area; one of these was plane, and the other very slightly curved, to cause the spark to pass always at a definite place. The electrical distribution on the opposing surfaces can be found (see above, Math. Theory of Electrical Equilibrium), as if the plates were plane and of infinite extent. This distance between the plates was measured by a micrometer, the contact reading being determined by observing when the electricity ceased to pass between the plates in the form of a spark. The potentials were measured in absolute electrostatic (C.G.S.) units, by means of Thomson’s absolute electrometer (see art. Electrometer). The limiting tension or dielectric strength is given in each case in grammes per centimetre, the formula for calculating it being

V 2 p- 8ιr×981∙4d ’ '

in which V represents the potential difference or electromotive force between the plates, and d the distance in centimetres. If we take the older view of Poisson’s time that the action of the electricity on the surface of a conductor is simply a fluid pressure, then p represents that pressure.

If we could consider the air between the plates as a homogeneous dielectric, then, for air at a given pressure (and temperature?) and given state of dryness, p, which measures its dielectric strength, would have a constant value independent of the distance between .the plates, and V would be proportional to d. A glance at Sir Wm. Thomson’s tables shows that this is not the case. For a distance of Ό0254 cm., p= 11 , 290, whereas for a distance •1524, p = - 535. It appears, therefore, that the dielectric strength of a thin stratum of air is much greater than that of a thick one. It is very difficult to understand why this should be so. “Is it possible that the air very near to the surface of dense bodies is condensed, so as to become a better insulator; or does the potential of an electrified conductor differ from that of the air in contact with it, by a quantity having a maximum value just before discharge, so that the observed difference of potential of the conductore is in every case greater than the difference of potentials on the two sides of the stratum of airby a constant quantity equivalent to the addition of about Ό05 of an inch to the thickness of the stratum!” It is remarkable that the limiting tension should be so small, somewhere about half a gramme per sq. cm., as compared with the atmospheric pressure, which is about 1032 gm. per jq. cm.

A series of absolute measurements of the potential required to produce a spark between equal spheres at different distances has beeu made by Mascart. The method employed was very ingenious.

Effect of Pressure, Temperature, <j ’ c∙., on the Dielectric • Strength of Gases.—The dielectric strength of a given gas depends on its pressure, or at all events on its density. Harris, who experimented on this subject, inclosed two balls in a receiver which could be exhausted to any required degree, and connected them with the armatures of a battery of jars. He found that the charge which had to be given to the battery in order to produce a spark between the balls was proportional to the density of the air in the receiver, while it seemed to be independent of its temperature. This amounts to asserting that the difference of potentials required to produce a spark between the balls is proportional to the density of the gas and independent of its temperature. Since we keep the distance between the balls the same throughout, this statement is equivalent to saying that the dielectric strength of a gas varies directly as its density, and does not depend on the temperature. Masson, using the method which Faraday had employed in comparing the dielectric strength of gases (vide infra) arrived at the same conclusion as Harris. Knochenhauer, however, experimenting with pressures ranging from 3 to 27 ’ 4 inches of mercury, found that for a given interval the difference of potentials required to produce disruptive discharge was proportional to the pressure increased by a small constant quantity.

Faraday, in the 12th and 13th series of his Experimental Researches, examines this subject; and the reader who desires to have a clear idea of what the issues involved really are will do well to begin by carefully studying Faraday’s results, and still more his views on this matter. Faraday directs his attention to the specific behaviour of different gases.

The gas to be examined was introduced into a receiver in which were arranged two balls s and l, of diameters 0 ’ 93 in. and 2‘02 in. respectively, at a constant distance 0 - 62 in. apart. Two balls, S and L, of diameters 0 - 96 in. and l - 95 in., were placed on suitable insulating supports outside the receiver. S and s were connected with an electric machine, and I and L to earth. The distance u between S and L could be varied at will; if it was greater than a certain value β, the sparks always passed between s and l in the receiver; if it was less than a certain value α, they always passed between S and L in the outer air. It might have been expected that α and ø would be equal, or at least very nearly so, i.e. that there would be one definite value of u, for which the spark would hesitate between the alternative intervals. This is not so, however. Nor again is the value of u the same when s and l are negative as when they are positive. The following table will illustrate these points, as well as the relations of the diiferent gases :— [table]

It will be seen that the different gases present considerable variety, and cannot be classified in any way so as to connect the dielectric strength with any other physical property. The numbers given cannot be regarded as measuring the dielectric strength, owing to the disturbing influences which cause the inequality of α and β. This inequality is not by any means small; e.g., for air the uncertainty amounts to about 32 per cent. These experiments show very clearly that the sign of electrification of the surface at which the discharge begins has a great effect on the limiting tension. The discharge passes much more readily from a small ball to a large one when the former is negative than when it is positive. Faraday made a variety of experiments to elucidate this point, and he was driven to the conclusion “that, when two equal small conducting surfaces equally placed in air are electrified, the one positively the other negatively, that which is negative can discharge to the air at a tension a little lower than that required for the positive surface, and that, when discharge does take place, much more passes at each time from the positive than from the negative surface.”

The inequality of α and β may be due to various causes, among which may be mentioned the charging of the glass of the receiver, dust, &c., in the air, heating of the air, and the presence of finely divided metal dispersed by preceding sparks. The last of these causes would account to a considerable extent for the fact that the sparks show a tendency to persist in a path once opened, and that the interval β - α is less for the negative spark, which starts at a smaller limiting tension, and may therefore be supposed to produce less mechanical effect.

Wiedemann and Rühlmann have recently taken up this subject in a research which has already been alluded to.

The gas and the spark terminals were inclosed in a cylindrical metal receiver with rounded ends. A small window allowed the light from the spark to fall on a rotating mirror fixed on the axis ofa HoΓtz machine, which furnished the electricity. The images of the successive sparks were observed by means of a heliometer. One-half of the divided object-glass was moved until one of the images of - one discharge coincided with one of the images of the next; then a similar coincidence was brought about by displacing the half-lens in the opposite direction. The difference (y) of the two readings on the micrometer of the heliometer measures the rotation of the disc of the Holtz machine between the two sparks. Preliminary experiments showed that the amount of electricity furnished by the machine while the disc moves through a given angle is independent of the angular velocity of the disc. It varies from day to day, however, according to the quantity of moisture in the air and the arrangement of the machine; but, on the principle just laid down, correction can easily be made by taking the reading each day of a galvanometer through which the current of the machine is sent. It follows, therefore, that y is proportional to the quantity of electricity which passes at each discharge through the gas, and by means of a galvanometer observations on different days can be compared.

It was found that at the lowest pressures worked with (∙5 to - 25 mm. of mercury) the discharge of the Holtz machine was still discontinuous; and that in all the experiments the tension at the electrodes was such that tho discharge was independent of the nature of the metal,—in other words, that the disintegration of the electrode played no essential part in the discharge.

The quantity of electricity required to effect a discharge, other things being equal, increases with increasing pressure. This increase is at first rapid, then slower, and at high pressures it is nearly proportional to the increase of pressure. It was found that y could be expressed with sufficient accuracy in terms of the pressure p by the empirical formula, y = A + Bp - C∕> 2 , in which the constants A, B, C depend on the size and insulation of the electrodes, their distance apart, and so on.

They arrange the gases in the following order of dielectric strength :—hydrogen, oxygen, carbonic acid, air, nitrogen. It is not a little remarkable that this is the order given by Faraday in the second column (the best) of the results we quoted above.

They find, in agreement with Faraday, that a greater quantity of electricity is required to bring two unequal spheres to the discharging point when the small one is positive than when it is negative. When two equal spheres are used, the value of y is least when both are insulated, greater when the positive sphere is uninsulated, and very much greater when the negative one is uninsulated.

All this is in accordance with theory, provided we assume with Faraday that the limiting tension is greater at positive than at negative surfaces. For example, suppose the surface densities corresponding to the limiting positive and negative tensions to be P and N (P≥N), and consider the case of two equal spheres of radius a, at so ∕α∖ 8 great a distance c apart that ( - I may be neglected, then by taking three consecutive images the reader will easily find that the charges which must be given to either ball in the case where both spheres are insulated and equally charged, and to the negative ball in the case where the positive ball is uninsulated, and to the positive ball when the negative ball is uninsulated, must be - 3 % 2 ^4πα 2 N, 4π∙α 2 N , 4πα 2 P, respectively, in order to produce discharge. The discharge begins at the negative ball in the first two cases, and at the positive ball in the third, and the quantities are obviously in ascending order of magnitude when P is ≥N.

The dielectric strength goes on increasing when the pressure is raised above the atmospheric pressure. Cailletet found that a powerful induction coil worked by eight large Bunsen cells was powerless to effect discharges across 1, mm. of dry gas at a pressure of 40 or 50 atmospheres.

On the other hand, however, the dielectric strength does not diminish indefinitely as the pressure decreases, but reaches a minimum.

Morren and De la Pdve have sought to determine this minimum dielectric strength by measuring by means of a galvanometer the mean intensity of the current sent through the gas by an inductorium so arranged that only the direct induction current passes; they thus obtain what they call a minimum resistance. Morren gives the pressures corresponding to this minimum for various gases; they lie between 0 - l and 3 - 0 mm. It may be questioned whether any very definite meaning can be attached to results of this kind; for the discharge is discontinuous, and resistance in the proper sense of the term cannot be spoken of.

It is clear, however, that a minimum dielectric strength must exist; for, if we go on improving our vacuum, we find that our ordinary machinery fails to send electricity through any considerable length of the exhausted space.

Morgan seems to have been the first to discover that the electric spark would not pass in a vacuum. Having carefully boiled the mercury in a barometer tube, so as to remove the last traces of moisture, he found that the inductive discharge caused by electrifying a piece of tinfoil on the outside of the tube would no longer pass to the mercury, and cause the luminous phenomena usually seen under such circumstances. Masson repeated this experiment in a more satisfactory form. Gassiot greatly improved the exhaustion of vacuum tubes by filling them with CO 2 , pumping out as usual, and then absorbing the residual gas by fusing a piece of KHO previously inserted into the tube. He constructed tubes in this way which had sufficient dielectric strength to insulate the pole of his great battery of more than 3500 Zn. Aq. Cm cells. Hittorf an<l Geissler have constructed vacuum tubes (by pumping with a Geissler’s pump, and heating the whole to 400° to 500° C.) in which the opposition to the discharge of an interval of ⅜ mm. between two platinum electrodes was greater than that offered by 15 or 20 centimetres of ordinary air.

Different Forms of the Discharge in Gases.—We have said that the subsequent progress of the disruptive discharge when once begun is influenced by a great variety of circumstances. The beginning of the discharge evolves heat, which rarefies the neighbouring air, and therefore weakens its dielectric strength. Owing to this cause the discharge once started tends to go on. Again, if any considerable quantity of electricity escapes into the ruptured dielectric at the first burst, this relieves the tension at the surface of the conductor. On the other hand, the progress of part of the electricity towards the opposing conductor raises the tension at the surface of the latter, so that disruptive discharge is provoked or helped there. If the initial tension is considerable, or the quantity of electricity which passes to begin with very great, glowing metal particles are shot forth into the dielectric, causing a reduction of its strength, which will be very different in different directions. Motions of the air play a great if not a preponderating part in many forms of the discharge. The electrification, &c., of the walls of the tube, and the form of the electrodes and of the tube, both in the neighbourhood of the electrodes and at a distance from them, are as important in their influence on the continuance of the discharge as they are on its start. And, last but not least, much depends on the way the electricity which produces the discharge is furnished,—on the nature of the electromotor, in short. Although we have not yet exhausted the influencing conditions, we have probably said enough to convince the reader that little aid is to be hoped for in this matter from considerations a priori. There is a great deficiency even in proximate principles to guide us in the maze of experimental detail; and although most of the experiments are beautiful beyond all conception, yet the mere narration w ould scarcely interest the reader. Our description of the department will, therefore, consist simply in going round the boundary.

The luminous appearances may be roughly classed under the forms of spark, brush, glow and convective discharge, and dark discharge.

At the ordinary atmospheric pressure the disruptive discharge between two conductors at a moderate distance apart takes place in the form of a brilliant sharply-bounded streak of light, whose apparent breadth is in general small. For small distances the spark is straight, and has the appearance of being thicker, or at least more brilliant, at the ends than in the middle. When the distance is considerably increased the spark assumes the characteristic zig-zag form seen in forked lightning. It seems occasionally to be absolutely broken by perfectly dark spaces. The duration of the discharge in this form, more especially when the resistance of the discharging circuit is very small, as tested by a rotating mirror, appears to be exceedingly short.

We have taken photographs of the sparks of a Holtz’s machine by simply moving the camera containing the sensitized plate vertically upwards past the electrodes of the machine. The result is a column of perfect photographs, quite unblurred by the jarring, &c., of the camera stand. Again, if a disc painted with white and t>lack sectors be caused to rotate very rapidly, it appears in ordinary light to have a uniform grey colour; but when it is viewed by the light of an electric spark the sectors are seen exactly as if the disc were at rest, which proves that the illumination lasts for a very short time. Masson founded on this experiment a beautiful method for measuring the intensity of the light given out by the spark. A description of his apparatus, with an account of his results, will be found in Mascart. I

The colour of the spark in air is bluish, but at the same time its great brilliancy gives an impression of whiteness. In nitrogen the appearance is much as in air, only the colour tends more to bluish purple, and the spark is more sonorous. In oxygen the spark is whiter and less brilliant than in air; in hydrogen crimson-coloured; in carbonic acid greenish; in hydrochloric acid white, and never broken by dark parts; in coal gas green or red, with occasional dark parts. If the spark be carefully examined, especially when the pressure is greater than an atmosphere, it will be seen that the central bright streak is surrounded by an envelope, of somewhat nebulous form, and of a lavenderblue colour. This envelope tends to spread over the negative electrode, where it is more conspicuous as compared with the central streak than elsewhere. This envelope appears to be due to the glowing metal particles torn from the electrodes. It has, unlike the central streak, a sensible duration, on account of which it happens in many cases that a much greater quantity of electricity passes through it than through the infinitely more brilliant but less enduring part of the discharge. The envelope can be actually separated from the streak by a current of air properly directed, or by the action of a magnet (vide infra, p. ~O∙

When the discharge in air at the atmospheric pressure takes place between a salient but not pointed part of one conductor and another conductor of considerable surface (e.g. between one sphere 2 cm. diameter and another 13 cm. diameter), the luminous appearance very often takes a characteristic form, which has been called the brush discharge. The name is to a considerable extent descriptive of the phenomenon; if the word broom had been applied it would have been even more appropriate, and a rough idea of the variety of forms the brush may assume will be obtained by thinking of the various forms of the domestic article in question. At the surface of the smaller conductor appears a short, straight, luminous stem differing in appearance very little except in brightness from a spark. From this radiate a series of twig-like branches of much inferior brilliancy, having a purplish-violet colour. These subdivide in many cases into still smaller ramifications, and are ultimately lost in the medium. When the large conductor is either altogether absent or very distant, the general tendency of the branches is to spread outwards more and more in all directions; but when the large conductor is brought nearer, the branches have a tendency to bend down towards it, so that the whole assumes an ovoid shape. The brush is generally accompanied by a crackling or hissing sound, or even a musical note. On approaching the hand or a conductor of extended surface, the pitch of this sound rises considerably. This at once suggests that the brush is an intermittent phenomenon. That this really is so was clearly proved by Wheatstone in one of the earlier applications of his rotating mirror. Wheatstone saw in his mirror not one image of the brush, but several arranged in succession at regular intervals. Each of these images corresponds to a single discharge, and each appears less complicated than the brush as viewed by the unaided eye, which is, in reality, a superposition of a considerable number of brushes, the number depending on the time taken by a light impression to fade on the retina. At the same time each individual image is a little drawn out in the direction of motion of the mirror, which shows that the brush has a sensible duration. Faraday speculates very acutely concerning the nature of the brush discharge (see Exp. Res., 1425 sqqj). He finds that, although it is generally accompanied by a current of air, yet it is not always or necessarily so. He also carefully illustrates the difference between the positive and negative brush. If we have a small ball on the end of a wire projecting freely into the air, the positive brushes obtained from it are much larger and finer than the negative brushes so obtained. Again, if we charge a large metal ball positively, and bring an uninsulated metal point up to it, a star appears on the point, which gets brighter and brighter as the point approaches the sphere, but the form does not change until the distance is very small. If the sphere be charged negatively, the star appears as before when the distance is considerable, but at a mode rate distance (1 to 2 inches) a brush forms, and when the distance is still farther reduced a spark passes. It seems, therefore, that the negative discharge keeps its form unchanged under considerable variety of influencing circumstances, whereas the form of the positive discharge is more readily affected. The explanation of these differences he finds in the fact, which he established by experiments already alluded to, that the limiting tension is smaller at positive than at negative surfaces; so that, cceterιs paribus, the negative discharge occurs of tener than the positive discharge; but, on the other hand, when the latter does occur, more electricity passes. This, no doubt, accounts for the lower pitch of the sound of the negative brush, and the greater extent and brilliancy of the positive one. Faraday found great differences in the character of the brush in different gases; in none apparently does it reach the brilliancy attained in air or nitrogen. He also observed that rarefaction up to a certain point favoured the production of brushes.

When discharge takes place from the rounded end of a wire projecting freely into the air, the brush is very often replaced by a quiet phosphorescent glow, which covers a greater or less extent of the end of the wire. The noise which accompanies the brush is entirely absent in this form of the discharge, and the means by which the brush can be analysed into a series of successive discharges give no corresponding result for the glow. In the rotating mirror it simply stretches out into a uniform band of light. The glow is therefore either a continuous discharge or an intermittent discharge of incomparably shorter period than the brush. Diminishing the discharging surfaces favours the production of glow. Increase of power in the electric machine which is furnishing the electricity has a similar effect. Rarefaction of the air has also a great effect in facilitating the production of glow, especially in the case of negative glow, which is extremely hard to produce in air at common pressures. In Faraday’s opinion, the star which is obtained with a positive sharp point is a positive glow; but he thinks it not improbable that the negative star is not a negative glow, but a small negative brush. The glow is invariably associated with a current of air to or from (generally both) the glowing conductor. Everything that favours this air-current increases the glow; e.g., a brush may sometimes be converted into a glow by properly directing an air-current near it. Again, everything that prevents or retards the formation of an air-current has a similar effect on the glow: a glow can be converted into a brush in this way. Lastly, everything which tends to prevent abrupt variation of the tension favours the glow, and everything having an opposite tendency is destructive of it. Faraday concludes, therefore, that the glow is due to a gradual discharge by convection, in which the agents are the particles of the gas. The order of the appearance of spark, brush, and glow at positive and negative surfaces is, in general, the same; but the gradation is different. Positive spark does not pass int» brush so soon as negative spark does; but, on the other hand, positive brush turns to glow long before negative brush.

Intimately connected with the glow is the convective discharge, if indeed they are not degrees of the same phenomenon. “The electric glow is produced by the constant passage of electricity through a small portion of air in which the tension is very high, so as to charge the surrounding particles of air which are continually swept off by the electric wind, which is an essential part of the phenomenon.” Now there seems little reason to doubt that at lower tensions discharge of this kind may occur without the luminous phenomenon at the surface of the conductor. If this be so, then the convective discharge is only a different degree of the glow discharge.

Discharge by convection plays a very important part in all electrostatical experimenting. The air in the neighbourhood of an electrified conductor gets charged, forming an electrical atmosphere, which surrounds the conductor, being more extensive in the neighbourhood of salient angles than elsewhere. Such electrical atmospheres are often a source of great inconvenience in the laboratory and lecture-room when delicate electrical experiments are in progress.

A curious little instrument, called the electrical tourniquet or windmill, depends for its action on the electrical wind which accompanies convective discharge. A small rectangular cross, with equal arms, is made of light wire; the extremities of the arms are bent through a right angle in the plane of the cross, so as to point all one way. The little cross thus made is poised, like a compass needle, on a vertical wire connected with an electrified conductor. Convective discharge takes place at the points, giving rise to an electrical wind, the reaction of which causes the little machine to revolve with great rapidity. If the experiment be conducted in the dark, a glow usually appears on the revolving points. The experiment also succeeds when the cross is immersed in a non-conducting liquid.

We have already alluded to the dark spaces that sometimes appear in the spark in gas at the atmospheric pressure. Faraday observed that a phenomenon of this kind was very common in coal gaβ. When the discharge takes place in highly rarefied gas, a dark space of this kind almost always separates the positive from the negative light, its situation having a certain degree of fixity with respect to the negative, but not to the positive electrode. It is very difficult to form an idea of the exact nature of the discharge which takes place in this space. Discharge there undoubtedly is of some kind; and pending further investigation, Faraday called it the dark discharge. The fact that its real nature is still undiscovered amply justifies the separate name. Faraday found that it occurred in discharges that pass almost instantaneously, aud concluded that it could hardly be due to convection of the ordinary kind, which requires time. De la Rive and Hittorf have made out many peculiarities connected with its appearance in vacuum tubes, the phenomena in which we now attempt briefly to describe.

A variety of forms may be given to the vessel in which the rarefied gas to be experimented on is inclosed.

One of the most common used to be the electric egg, which is simply an oval glass vessel furnished with two small metal spheres for electrodes; the stems which carry these electrodes pass air-tight through tubes cemented to the ends of the vessel; the stem which supports the whole is perforated and fitted with a stop-cock, so that the apparatus can be exhausted to any required extent and then temporarily closed. The commonest of all instruments of this kind now-a-days is the Geissler tube This is simply a glass tube, into which are fused two electrodes of platinum or other metal; a capillary tube allows the apparatus to be connected with an air-pump, and exhausted; when this is done, the capillary tube is sealed up by means of a spirit-lamp. A very common form of such tube is the spectrum tube (see art. Light), consisting of two wider parts, connected by a capillary part, in whictι the light of the discharge is much more intense than elsewhere. Complicated tubes of all kinds have also been constructed as electric toys.

The reader must not forget that the form of the tube exercises a great influence on the phenomena, whether at the positive or negative electrode. In the summary description that follows the electric egg is referred to, unless it is otherwise stated. We further assume that the electromotor used gives currents in one direction only. A Holtz machine would satisfy this condition, within certain limits at least.

When the gas is rarefied to a considerable extent, the spark loses its sharp outline, becomes interspersed with nebulous portions, and by-and-by loses its characteristic form altogether. As the rarefaction goes on, the discharge ceases to reach from the positive to the negative electrode. The latter now displays a patch of lavender-blue light, separated from the positive light by a dark interval, the length of which depends on the distance between the electrodes. In certain cases the positive light terminates in a cup-shaped depression, whose concavity is turned towards the negative electrode. As the rarefaction is still further increased, the positive light tends more and more to fill the tube, although in general it recedes from the negative electrode, over which, on the other hand, the beautiful lavender glow spreads more and more, exhibiting at the same time a growing tendency to fill a limited space surrounding the electrode. At a still higher degree of rarefaction, the positive light, which now occupies a considerable space, and takes a shape more or less corresponding to that of the inclosing vessel, is divided transversely iuto a number of cup-shaped striae, separated from each other by darker intervals. These striae vary in form and appearance considerably, according to circumstances. In the neighbourhood of the positive electrode, their concavity is turned towards the positive electrode; but towards the other end of the positive light, the concavity may be turned the other way, especially in the electric egg. The positive light, in vacuum tubes, shows therefore the same remarkable variability, and the negative light the same measure of stability that Faraday remarked in gas at ordinary pressures. The colour of the positive light varies very much in different gases; in nitrogen and air its rosy-red colour contrasts very sharply with the blue of the negative light. The negative light is remarkable for its power of producing fluorescence. It is very dependent as to its extent on the form and size of the uncovered surface of the electrode; anything placed on the electrode cuts it off sharply, as if the light were projected from the electrode and stopped by the obstacle. Disintegration of the negative electrode also goes on very rapidly, so that, after a vacuum tube has been used for some time the glass all round the negative electrode is blackened, browned, Ac., as the case may be, with a deposit of finely divided metal. The quantity as well as the quality of this deposit depends very much on the nature of the metal; it is smallest with aluminium, which is on that account much used for electrode terminals. The negative light occasionally shows one, two, or even three stratifications; but in this respect it never equals the positive light. When the rarefaction is carried to the utmost, both positive and negative lights fall off greatly in splendour. The negative light contracts more and more in upon the electrode, and confines itself even there to a small patch near the end, showing, however, a tendency to pass along the axis of the tube towards the positive electrode. The positive light, on the other hand, gradually draws inwards, till at last it is only a star on the end of the electrode, which now disintegrates, owing to the great tension.

The temperature at the two electrodes is, in general, very different. The true explanation of this difference has not been made out, although it is doubtless connected with the equally unexplained differences in the light phenomena. A general rule has been laid down, that the temperature of the negative electrode is always higher when the discharge takes place through the gas alone, and the temperature of the positive electrode higher when the discharges pass mainly through particles of disintegrated metal. The former case is commoner in vacuum tubes, where the negative electrode may get white hot, and even melt, while the positive electrode remains quite dark. The latter case is exemplified in the voltaic arc, in which great disintegration of the positive electrode is accompanied by a higher temperature there. Attempts have been made to investigate the temperature in different parts of the tube, and it seems to have been made out that the temperature is lower in the dark intervals than elsewhere.

When the electromotor is an induction coil, which furnishes discharges alternately in opposite directions, there will be a mixture of positive and negative light at each electrode, unless the maximum tension corresponding to the inverse discharge be so small that the direct discharge alone can break through. If, however, the tube be examined by means of a rotating mirror, or if it be itself fastened to a rotating arm, the images of the different discharges will be separated, and it will be seen that the appearances at each electrode alternate.

Again, when a Leyden jar is discharged through a vacuum tube, the appearances at the two electrodes are often very much alike, particularly when the resistance of the discharging circuit is very small. When the resistance is increased by introducing a column of water or lengths of wetted string, the appearances are similar to those indicated in our summary description. The reason of this is fully explained by the observations of Feddersen. He examined the spark of a Leyden jar by means of a rotating concave mirror. The machine which drove the mirror had a contact-maker, which brought on the discharge when the mirror was at a definite position; the image of the spark was thus thrown by the mirror on a piece of ground glass or a photographic plate, properly placed to receive it. He found that the discharge assumed three distinct characters as the resistance of the discharging circuit was gradually decreased.

1. The discharge was intermittent, that is to say, consisted of a series of partial discharges all in the same direction, following each other at more or less irregular intervals.

2. When the resistance was reduced to a certain extent, the discharge became continuous. The image of the spark on the plate had then the form of an initial vertical strip, with two horizontal strips extending from each end, and gradually thinning off to a point. The vertical strip indicates a single initial spark, and the horizontal bands the finite duration of the light from the glowing metal particles, &c., near the electrodes.

3. When the resistance is very small, the discharge is oscillatory, i.e., consists of a succession of discharges alternately in opposite directions. These oscillations are due to the self-induction of the discharging circuit; we shall examine the matter more carefully under Electromagnetic Induction.

It is obvious that when the discharge is either intermittent or continuous, the luminous phenomena will be of the normal form sketched above, but when the discharge is oscillatory there will be a mixture of positive and negative appearances at each electrode, the independent existence of which cannot be detected by the unaided eye.

This is the place to remark that it is rarely that the discharge is of the simple form (2), t.e., consists of a single continuous discharge; in by far the great majority of cases it consists of a series of partial discharges. With the inductorium, both varieties (1) and (3) may occur according to the length of the air space, the resistance of the whole secondary circuit, and so on. A number of very beautiful experiments have been made to illustrate these principles, which it would take us beyond our limits to describe. Good summaries of the results of Felici, Cazin and Lucas, Donders and Nyland, Ogden Rood and Alf. Mayer, will be found in Mascart and Wiedemann. Recent researches of a very important character have been made by Wiillner @ 1 and Spottiswoo'de on the discharge in vacuum tubes. They employ the rotating mirror. It would be premature to attempt to sum up or criticise their results, suffice it to say that they show an amount of agreement which augurs well for the future of this branch of electrical science. The striae seem, according to them, to play a more essential part in the phenomenon than was perhaps previously expected. Spottiswoode, in fact, seems to incline to the view that *11 discharges having a dark interval are really stratified, although, owing to their rapid motion, the strata may not be distinguishable by the eye alone.

In connection with this subject it may be well to mention the early experiments of Wheatstone, to determine the so-called velocity of electricity in conducting circuits. Six balls, 1, 2, 3, 4, 5, 6, were arranged in a straight line on a board; 2 and 5 were connected with the coatings of a charged Leyden jar; discharge passed by spark from 2 to 1, then through a large metallic resistance to 3, thence by spark to 4, then through a large metallic resistance to 6, and thence by spark to 5. It was found, as Feddersen observed later, that the introduction of the metallic resistance increased the duration of the sparks at all the intervals, so that the images in the mirror were ZZnew of small leugth; but, in addition, the spark between 3 and 4 began a little later than the sparks at 1, 2 and 5, 6, which were simultaneous. From this the velocity of electricity has been calculated, by taking the interval between the sparks to be the time which the electricity takes to travel through the metal wire between the intervals. Faraday long ago pointed out that this interval depends on the capacity of the wire, and may vary very much according to circumstances. It is very great in submarine telegraph wires for instance (vide supra, p. 36). Accordingly, the values of the so-called velocity of electricity, which have been found by different observers, differ extremely.

The sketch we have just given of the disruptive discharge in rarefied gases must be regarded as the merest outline. There are many points of great importance to which we have not even alluded. Hittorf’s investigation on what has been called the “resistance” of different parts of a vacuum tube during the discharge has not been mentioned, although it led to results of much interest, which must come to be of great importance when the clue to an explanation of the whole phenomena has been found. The reader who desires to study the matter will find in Wiedemann an excellent account of Hittorf’s work, with references to the original sources. We have not so much as raised the delicate and difficult questions concerning the spectroscopic characteristics of the discharge. A good part of this subject belongs indeed more properly to the science of Light.

Miscellaneous Efects, chiefly Mechanical.—Owing to the heat suddenly developed by the electric spark, and perhaps to a specific mechanical effect as well, there is a sudden dispersion in all directions of the particles of the dielectric. This commotion may be shown very well by means of Kinnersley’s older form of the thermo-electrometer; or Gauss’s instrument may be used if we replace the thin wire by a couple of spark terminals. When the spark passes, the liquid in the stem sinks suddenly through a considerable distance, even if the spark be of no great length (2 to 3 mm.).

Very curious effects are obtained when an electric spark is repeated several times at a little distance above a plate strewed with finely powdered chalk. After a time the chalk is seen to be divided by a network of fine lines, resembling the markings on shagreen. If α plate of glass be covered with powdered charcoal, and the spark passed through the powder, it arranges itself in a series of striae closely resembling those seen in a vacuum tube.

The power of the spark to induce chemical combination (in particular, combustion) is due no doubt mainly to its high temperature.

The discharge through non-conducting liquids may take place in the form of spark or brush. The brush, however, is poor compared with that obtained in air, and is very hard to get. When the spark passes, pressure is suddenly transmitted through the fluid in all directions, and if it be inclosed in a tube the tube is generally broken, even when the spark is by no means long. When the surface of the liquid is free, a considerable portion is usually projected into the air. The convective discharge is very marked in liquids. If two small balls connected with the electrodes of a Holtz’s machine in action be dipped in paraffin oil at a small distance apart, the whole liquid is thrown into violent motion by the convection currents, runs up the wires which lead to the balls, and spouts off in little jets.

There is also a distinct heaping up of the liquid between the balls, and if one of them be gradually withdrawn from the liquid, for a centimetre or so it raises a column after it, which adheres until the machine is stopped. It is very probable that other effects due to the alteration of the apparent surface tension, owing to the difference of electrical stress in the air and oil, are present in these phenomena, but this is hardly the place to discuss the matter.

The electric discharge passes with great facility through card-board and other bodies of loose texture. In all probability the air in such cases has quite as much to do with the resulting effects as the solid body.

A curious experiment of this kind is often made. Two points are arranged so as to touch the opposite sides of a piece of card-board. If the points be opposite each other, the discharge passes straight through, leaving in the case of small charges a tiny hole with burnt edges. If, however, the points be not opposite each other, the perforation occurs in the neighbourhood of the negative point. The peculiarity is no doubt connected with those differences between positive and negative discharges in air which we have several times noticed above. In fact, it is found that in an exhausted receiver the card is pierced at a spot very nearly equidistant from the two points. .

In other cases the main part of the dielectric strength depends on the solid material. The power of such bodies to sustain the electrical tension is often very considerable. Yet there is a limit at which they give way. A thickness of 6 centimetres of glass has been pierced by means of a powerful induction coil.

In such experiments special precautions have to be taken to prevent the spark from gliding over the surface of the glass instead of going through; this is managed in some cases by embedding the glass along with the terminals of the coil in an electrical cement of considerable insulating power; in ordinary experiments, however, it is in general sufficient to place a drop of olive oil round one of the terminals where it abuts on the glass. The appearance of the perforations depends considerably on the quantity of electricity that passes in the discharge. In some cases the glass cracks or even breaks in pieces. In some large blocks we have seen a perforation in the form of several independent threads, each of which had a sort of beaded structure, which may possibly be in some way analogous to the stratifications in vacuum tubes.

' Discharge along the Surface of a Body, Dust Figures, and Dust Images.—The class of phenomena referred to under this head are remarkable for the methods by which they are usually demonstrated. They were at one time much studied on account of the light they were supposed to throw on the nature of the so-called electric fluid or fluids. Though no longer regarded in this light, they have reference to an extremely important and comparatively little studied subject, viz., the distribution of electricity over the surface of non-conductors. It is easy to see that the demonstration of surface electrification on insulators is beset with difficulties of a peculiar kind. A very convenient method is to project on the surface a powder electrified in a known way; this powder clings to the parts oppositely electrified to itself, and avoids those similarly electrified, so that the state of the surface is seen at once. Lycopodium seed and powdered resin have been used in this way; they are sifted through linen cloth, the lycopodium becoming thereby weakly positive, and the powdered resin strongly negative. If the lycopodium be used, it covers both positive and negatively electrified patches, only the latter more thickly than the former.

The powdered resin, on the other hand, covers the positive and avoids the negative regions. The most effective powder, however, is a mixture of flowers of sulphur and red lead. In the process of sifting, the red lead powder becomes positively and the sulphur negatively electrified, and the powders separate themselves. The sulphur colours positive regions yellow, and the red lead colours negative regions red. The result is very striking; and the test is found to be very delicate.

The dust figures of Lichtenberg are one of the best known instances of the kind of experiment indicated above. A sharp-pointed needle is placed perpendicular to a non-conducting plate, with its point very near to or in contact with the plate. A Leyden jar is discharged into the needle, and the plate is then tested with the powder. If the electricity communicated to the needle was positive, a widely extending patch is seen on the plate, consisting of a dense nucleus, from which branches radiate in all directions. If negative electricity was used, the patch is much smaller and has a sharp circular boundary entirely devoid of branches. This difference between the positive and negative figures seems to depend on the presence of the air; for the difference tends to disappear when the experiment is conducted in vacuo. Riess explains it by the negative electrification of the plate caused by the friction of the water vapour, <fcc., driven along the surface by the explosion which accompanies the disruptive discharge at the point. This electrification would favour the spread of a positive, but hinder that of a negative discharge. There is, in all probability, a connection between this phenomenon and the peculiarities of positive and negative brush and other discharge in air; Riess, indeed, suggests an explanation of the latter somewhat similar to the above.

There is another class of figures, to which Riess gives the name of electric images, of which the following may be taken as a type. A signet or other engraved piece of metal is placed on a plate of insulating material, and steadily electrified by means of a dry pile or otherwise positively or negatively for half an hour or so. When the metal is removed and the plate dusted, an exact figure of the stamp appears, consisting of a red or yellow background on which the engraved lines stand out free from dust. There is no difference between positive and negative electricity here as far as form is concerned, and the colour of the figure indicates charge on the plate opposite to that on the metal. The phenomenon appears to be due simply to the electrification of parts of the non-conducting surface opposite the metal.

Another class of phenomena, to which Riess gives the name secondary, depend, not on the electrification of the surface, but on permanent alterations produced by the discharge, whether in the form of spark or otherwise. Sometimes these are directly visible to the eye or touch, e.g., the roughening and discoloration which mark the path of the spark over a polished glass surface. In some cases they are chemical alterations, which may be shown by means of the proper reagents, e.g., the separation of the potash in the spark traces on glass. In certain cases they become evident on breathing upon the glass; of this description are the images of Karsten. A piece of mirror glass is placed on an uninsulated metal plate, and on the glass is placed a coin or medal. Sparks are taken for some time between the coin and an electric machine, and then the glass plate is removed and breathed upon. A representation of the coin then appears on the glass, often complete to the smallest detail. The reader whois interested in these matters, historically or otherwise, will find a variety of information, with directions how to find more, in Riess’s Reibungsclcctrieitiit, Bd. ii. § 739 sqq.

Electromagnetism and Electrodynamics. ·

Mention has already been made of the discovery of Oer sted, that the electric current exerts a definite action on a magnetic needle placed in its neighbourhood. This discovery formed the starting-point of that division of electrical science with which we are now to deal. It was natural, once the action of a current on a magnet was observed, to look for the reaction of the magnet on the current, and after seeing two currents act on the same magnet, it was reasonable to expect that the currents would act on each other. Yet it may be doubted whether the first of these results is a legitimate deduction from the discovery of Oersted, and the second certainly is not so. Before we can apply the principle of the equality of action and reaction we must be quite certain of the source of the whole of any action to which the principle is to be applied. Again, two bodies A and C may act on B owing to properties acquired by virtue of B’s presence, so that in the absence of B they need not necessarily act on each other. A good example is the case of two pieces of perfectly soft iron, each of which will act on and be acted on by a magnet, but which will not act on each other when the magnet is not near them.

The questions thus raised by Oersted’s discovery were experimentally settled by Ampère. He found that a magnet or the earth (which behaves as if it were a magnet) acts oxi the current, and the direction of these actions is found to be consistent with the principle of equality of action and reaction. As no experimental fact has yet been quoted against the application of this principle in such cases, we shall assume it henceforth. Ampère also discovered the action of one electric current on another, and thereby settled the second question. We may conclude, therefore, that the space surrounding an electric current is a field of magnetic force just as much as the space around a magnetized body.

The next step is to determine the distribution of magnetic force, or what amounts to the same thing, to find a distribution of magnetism which shall be equivalent in its magnetic action to the electric current. This aLso was completely accomplished by Ampère. In expounding his results we shall follow the order of ideas given by Maxwell, which we think affords the simplest view of the matter, and is the best practical guide that we know of through the somewhat complicated relations to which the subject introduces us. We shall in addition give a sketch of the actual course which was followed by Ampère, and which is adhered to by the Continental writers of the present day.

It results alike from the fundamental experiments of Ampère and the elaborate researches of Weber, to both of which we shall afterwards allude, that an electric current circulating in a small plane closed circuit, acts and is acted upon magnetically exactly like a small magnet placed perpendicular to its plane at some point within it, provided the moment of the magnet be equal to the strength of the current multiplied by the area of the circuit, and its north pole be so placed that the direction of the axis of the magnet (from S-pole to N-pole), and the direction in which the current circulates are those of the translation and rotation of a right-handed (ordinary) screw which is being screwed in the direction of the axis. In this statement we have spoken of a small closed circuit. The word “small” means that the largest dimensions of the circuit must be infinitely smaller than its distance from the nearest magnet or electric current on which it acts, or by which it is acted on.

We may break up our small magnet into a number of similar magnets, and distribute them over the area of the small circuit, so that the sum of the moments of all the magnets on any portion w of the area is wz, where i is constant. We thus replace the circuit by a “magnetic shell” of strength i, which, if we choose, may be represented by two layers parallel to the area, one of north the other of south magnetism, the surface density of which is i÷θ, where b is the distance between the layers.

Starting from the principle thus laid down we eau derive all the laws of the mutual action of magnets and electric currents.

Consider any finite circuit ABC (fig. 29). Imagine it filled with a surface of any form, and a network of lines drawn on the surface as in the figure, dividing it up into portions, such as abed, so small that they may be regarded as plane. It is obvious that any current of strength i circulating in ABC may be replaced by a series of closed currents, each of strength i circulating in the meshes (such as abcdj of the network on the surface; for in each line such as bc we have two equal and opposite currents circulating whose action must be m7. Now, we may replace each of the small circuits by a magnet as above, or by a magnetic shell of strength i. The assemblage will constitute a magnetic shell of strength i filling up the circuit, whose magnetic action, at every point external to the shell will be the same as that of the current. The north side of the shell is derived from the direction of the current by the right-handed screw relation given above.

If <7S be an elemem of tne surface of a magnetic shell of strength i, D its distance from P, and θ the angle which the positive direction of magnetization (which is normal to <7S) makes with D. then the magnetic potential at P is given by

.∙ /7*cos 0

v =fV-^< is · · · w, the integration extending all over S.

When properly interpreted this double integral is found to represent the “solid angle” subtended at P by the surface S, or, as it may also be put, by the circuit ABC which bounds it. Hence, solid angles subtended by the north side being taken as positive, and the usual conventions as to sign adhered to, we may write

V = fω (2), where ω is the solid angle in question.

We see, therefore, that the potential of a magnetic shell at any point P is equal to the product of the strength of the shell into the solid angle subtended by its boundary at P. Now the potential of such a shell is continuous and single-valued at all points without it. (With points within it we are not now concerned, since the action of the current at such points is not the same as that of the shell.) If, therefore, a unit north pole start from any point P and return to the same, after describing any path which does not cut through the shell, i.e., does not embrace the current, the work done by it will be nil. Let us now examine what happens if the path cuts through the shell S. Take two points P and Q, infinitely near each other, but the one P on the positive side, the other Q on the negative side of the shell. In passing from P to Q, without cutting the shell, the solid angle ω decreases by 4π∙ infinitely nearly. Now, during the passage from Q to P we may not represent the action of the current by S, but nothing hinders us from representing its action by another shell S', which does not pass between Q and P, but is at a finite distance from either of them; for it will be remembered that the shell which represents the action of a current i is definite to this extent merely—that its strength is i, its boundary is the circuit, and it does not pass through the point at which the action is being considered. But infinitely little work, owing to the action of S', is done in passing from Q to P. Hence the work done by a unit pole in going once completely round any path which embraces the current once is 47«.

To reconcile this result with the continuity of the magnetic potential of a linear circuit, for the existence of which we have now furnished sufficient evidence, we must admit that the potential of a linear circuit at any point P is V = z(ω + 4∕i7r), where n is any integer. In other words, V is a many-valued function differing from i times the solid angle subtended at P by a multiple of 4τri. If we pass along any path from P and return thereto, the difference of the values of V, or the whole work done on the journey, is zero if the path does not embrace the circuit, 4nπ∙t if it embraces it n times

The considerations enable us to determine the action of any closed current on a magnetic pole, and consequently on any magnetic system. We have next to find the action on a linear circuit when plaeed in any given magnetic field, whether due to magnets or electric currents. This we do by replacing the circuit acted on by its equivalent magnetic shell.

I f the potential at any point of the magnetic field be V, then the potential energy of a magnetic shell S, of strength i, placed in the field is given by

, r .∕7Y,<*V d∖ τ dV∖ ...

il =⅛∕ l⅛ +,n ⅜^ + ”'3r) & ’ · · (8) ·

where (Z, m, η) are the direction cosines of the positive direction (south to north) of the normal to the element dS. Since, so long as the magnetic force considered is not due to S itself, there is none of the magnetism to which V is due on S, we may write - a, - b, - c for jfa ’ ^Ty ’ ^fi ’ where a, b, c are the components of the magnetic induction. Then, ii N = ff (ja + mb + τw)dS (i.e., =the surface integral of magnetic induction, or the number of line3 of magnetic force which pass through the circuit), we may write

M=-ιN (4).

From this expression for the potential energy of the equivalent magnetic shell we can derive at once the force tending to produce any displacement of the circuit regarded as rigid.

Thus let φ be one of the variables which determine the position of the system, then the force Φ tending to produce a displacement <lφ is given by Φdφ + dM = 0, or

Φ≈t 3 - (5).

dφ

Hence the work done during any displacement of a closed circuit, in which the current strength is i, is equal to i times the increase produced by the displacement in the number of lines of force passing through the circuit. The force tends, therefore, to produce the displacement or to resist it, according as the displacement tends to increase or to diminish the number of lines of force passing through the circuit. It is evident, therefore, that a position of stable equilibrium will be that in which the number of lines of magnetic force passing through the circuit is a maximum. If that number is a minimum, we have a case of unstable equilibrium.

Maxwell has shown how we may deduce from the above theory the force exerted on any portion of the circuit which is flexible or otherwise capable of motion. “If a portion of the circuit be flexible so that it may be displaced independently of the rest, we may make the edge of the shell capable of the same kind of displacement by cutting up the surface of the shell into a sufficient number of portions connected by flexible joints. Hence we conclude that, if by displacement of any portion of the circuit in a given direction the number of lines of induction which pass through the circuit can be increased, this displacement will be aided by tho electromagnetic force acting on the circuit.”

From these considerations we may find the electromagnetic force acting on any element ds of the circuit. Let PQ (fig. 30) ba the element ds belonging to the arc AB of any circuit. Let P$ be the direction of the magnetic induction at P, and $ its magnitude. It is obvicus that no motion of PQ in the plane of PQ and P$ will increase or diminish the number of lines of force passing through the circuit; consequently no work will be done in any such displacement. Hence the resultant electromagnetic force R must be perpendicular to the plane QPg. Let PR be a small displacement perpendicular to this plane, the work done in the displacement is R.PR, and the number of lines of force cut through is i times the rectangular area PQR multiplied by the component I) sin Ö of the magnetic induction perpendicular to it. Hence we have

R X PR=ι⅛ × PR × ÿsinô, i.e. R = A⅛⅜}sinβ (6).

Hence the resultant electromagnetic force on the element ds may be determined as follows :—Take in the direction of the resultant magnetic induction (magnetic force) and proportional to Z36, and take PQ in the direction of ds and proportional to it; the electromagnetic force on the element of the circuit is proportional to the area of the parallelogram whose adjacent sides are P∣3 and PQ, and is perpendicular to it. The force in any direction making an angle φ with the direction of the resultant is of course Rcosφ. The following consideration is convenient for determining which way the resultant force acts. It is obvious that the force on the element wdll be the same to whatever circuit we suppose it to belong, so long as the direction and strength of the current in it is the same. Take, then, a small circuit PQK perpendicular to the line ·: of magnetic induction (magnetic force) near PQ, in such a way that the direction of the current in PQK (as determined by the direction in PQ) is related to the direction of the magnetic induction in the same way as rotation and translation in right-handed screw motion; then the element PQ tends to move so that the number of lines of force passing through PQK increases.

Several other ways of remembering this direction might be given. Although the above may sound arbitrary and look clumsy at first, yet we have found it more convenient in practice than some others we have tried.

We may extend what has been said above to the case where part of the magnetic force, it may be the whole of it, is due to the current in the circuit itself; for we might suppose the magnetic field to be that due to a shell whose boundary coincides infinitely nearly with the circuit. If the circuit is rigid, there will of course be no motion caused by its own action; but if it be flexible, there may be relative motions; in fact each portion will move until the number of line3 of force that pass through the circuit is the greatest possible consistent with the geometrical conditions.

It is an obvious remark, after what has been said, that the potential energy of the magnetic shell which represents a current depends merely on its boundary, or, in other words, that the magnetic induction or the number of lines of magnetic force which pass through a circuit depends merely on its form. Hence we should expect to find some analytical expression for the surface integral of magnetic induction depending merely on the space relations of the circuit; in other words, we should expect to find a line integral to represent it. And when the field is that of another circuit, we should expect to find a double line integral for the mutual potential energy of the two representative shells. We shall describe briefly how these expectations are realized.

In the first place, a vector may be found which has the property that its line integral taken round any circuit is equal to the surface integral of magnetic induction taken over any surface bounded by the circuit. This vector has been called by Maxwell the “vector potential” (®). Let its components be F, G, H. Then applying the definition to small areas dydz, dzdx, dxdy, at the point xyz perpendicular to the three axes, a, b, c being components of magnetic induction as before, we get

_ <*H *G ⅛ 45- dπ dG-rfF

dy dz' dz dx , C dx dy " '

These equations might be used to determine F, G, H, and would lead to a much more general solution than is here required. The following synthetical solution is simpler.

Consider a magnetized particle sn at 0 (fig. 31). Let the positive direction of its axis be OK, and let its moment b∙ m. The resultant force due to sn at any point P is in a plane passing through OK; hence the vector potential H at P must be perpendicular to this plane. Let its direction be taken so as to indicate a rotation round OK, which with translation along OK would give right-handed screw motion. Describe a sphere with 0 as centre and OP ( = D) as radius. Let PQ be a small circle of this sphere whose pole is K. Consider the line integral round PQ, and the surface integral over the spherical segment PKQ. Since ⅞l is the same at all points of PQ by symmetry, the former is 2πDsinβ51, and the latter is

~~~Β Π ’ E< l uati, ’6 th ese we g°t for vector potential of in at P

β-gsinβ (8), its direction being that already indicated.

Suppose now the particle sn placed at Q(a∙yj) so that the direction cosines of sn are λ, μ, v. Let the coordinates of P be ξ, η, also let QP—D—+ √(ξ-αr) 2 + (tj -jf) 2 +(£-z)*. Then the direction cosines of QP are D s ⅛, D 8 ⅛ , D'-⅛, where p = g; and we get for the component of the vector potential at P

'-(*2-⅛) »

and two similar expressions for G and II.

The vector potential of a magnetized body may be got by compounding the vector potentials of the different elements; hence, Iλ, Ip, Iv being the components of magnetization at any point of the body, we get τ ~fff'{ μd P d f) dlcdy ' il < 10 >∙ and two similar expressions for G and II. The first part of our problem is thus solved.

Let us, in the second place, apply the above result (10) to the case of the two shells which are equivalent to two currents. In a lamellar distribution of magnetism & c .; hence the dz dy volume integral in (10) reduces to a surface integral, and

F=y∕∕^I(>-κm)√S .... (11), where l, m, n are the direction cosines of the outward normal to dS.

Now the magnetic shell of thickness τ and strength i is a lamellarly magnetized body of constant intensity∙=t÷τ. It may be looked upon as bounded by two parallel surfaces normal everywhere to the lines of magnetization, and by an edge generated by lines of magnetization. At every point on either of the parallel surfaces we have therefore Z = λ, m = μ, n —r; and at the edge l = v⅛l - μ j, and similarly for τn and n. Hence every element of the double integral in (11) belonging to either of the parallel surfaces vanishes, and there remain only the parts on the edge which give F≈⅛i < 12 >-

since + + v⅜ i =0. (12) gives the vector potential as as as at (ξvC) due to a magnetic shell S. Let be any point on the boundary of another shell S, of strength i, and let dσ be the element of arc of the boundary, then -∕( f s + g ⅜ +h ¾>∙ · · ' 13 > is the magnetic induction through S' due to S with the sign changed, in other words, the mutual potential energy Μ. Putting for F,G,H their values by (12), we have m -<d(⅛⅛÷⅛⅛÷⅛⅛H

- - ii 'Jf dsdσ (1 *)»

where e is the angle between ds and dσ.

The result of (14) realizes the second of our expectations. The double integral arrived at is of great importance, not only in the theory of electrodynamics, but also as we shall see in the theory o the induction of electric currents.

Hitherto we have spoken only of closed circuits, and considered merely the action of a circuit regarded as a whole. When we did speak of the force on an element of a circuit, we deduced this force directly from the state of the magnetic field in its immediate neighbourhood. There is an order of ideas, however, in which the mutual action of two circuits is considered to be the sum of all the mutual actions of every element in one circuit on every element in the other. Now, we can easily show, by means of (14), that a system of elementary forces of this kind can be found which will lead to the same result for closed circuits as the theory given above.

Let the circuit S' be supposed rigid and fixed, and let the circuit. S be movable in any way with respect to S'; it may even be flexible.

Denote the angles between the positive directions of dσ and ds and the direction of D from dσ to ds by 0' and 0, then we have n dD λ , dD s COSØ =—J—, COSØ = --y—, j ds dσ ∖ . . (15).

dD dD τx d 3 D (

C0Se ds dσ D dsd<r '

By means of these we get il - ii ffi 'r⅞ ’ ' w, ∙ · <“>

The part which is a complete differential has been left out, because it disappears when the integration is carried round closed circuits, as we always suppose it to be. Consider now the work done in a small displacement which alters D and S, c L~ — , and ds, but dσ ds not dσ∙, we have r∖γ √√'∕7*1 dD dD , rι , , , ∙√ ∕*∕"1 dδD dD . , 5 Μ=-..γ/___«^ +11 γχ η _**

+ *7Z i l W - '⅞t'⅞ i v⅛∙⅛

J J D dσ∖ ds ds ds J A-√√∕ r ∕ rl c * d dδs . 1 f/J D⅛ *- -.u d ∙ ,,li ·

The parts containing δs disappear in this expression, and if the rest be arranged by integration by parts as usual, we get δM-√∕RδDd.s∙dσ∙ = O (17), where R j ,y cos ⅜ - 3 cos 0 cos 0'

D i

Hence the electrodynamical action of the two circuits is completely accounted for by supposing every element dσ to attract QNQry element ds with a force ii'dsdσf ∖ . .

—∣y 4 —I 2 cos e - 3 cos 0 cos 0' I · · (1°)∙

We may therefore use this elementary formula whenever it suits our convenience to do so.

It is very easy to obtain a similar elementary formula, which is very often useful, for the action of an element of a circuit on a unit north pole.

We have seen above how to find the action on an element PQ (ds) of a circuit in a given magnetic field. Let the field be that due to a unit north pole N (fig. 32). Then the magnetic induction at P is in the direction NPK, and is equal to 1 , if NP = D. Hence by (6) the force R, on PQ is perpendicular to NP and PQ, is in the direction PM shown in the figure, and is equal to θ Now, by the princip of “action and reaction,” the force on N is R in the direction PM' opposite to PM, he. is equal to a force R acting at X in a direction NM"parallel to PM', together with a couple whose moment is R× PN, and whose axis is perpendicular to NP and in the plane NPQ. Now a simple calculation, which we leave to the reader, will show that for any closed circuit the resultant of all the couples thus introduced is nil; hence, since we deal with closed circuits only, we may neglect the couple.

The force exerted by a closed circuit on a unit north pole may therefore be found by supposing each element ds to act on the pole with a force equal to ids sin 0 (19),

D 2 whose direction is perpendicular to the plane containing the pole and the element, and such that it tends to cause rotation round the element related to the direction of the current in it by the right-handed screw relation.

Comparison of Theory with Experiment.—The best verification of the theory which has just been laid down consists in its uniform accordance with experience. We proceed to give a few instances of its application, adopting now one, now another, of the equivalent principles deduced from it.

We have already remarked that the lines of magnetic: force in an electric field due to an infinite straight current 1 are circles having the current for axis. It is easy to deduce from the fact that there is a magnetic potential that the force must vary inversely as the distance from the current.

This may also be proved by means of the formula (19); in fact, the resultant force at P is given by

„ . ∕*sin0 , . f κ sinø ,. 2i , n ..

K = · < 20 >∙

u d being the distance of P from the current.

Let AB (fig. 33) be a very long straight current, and poq an element ds of a parallel current, having the same direction as AB. If we draw the line of force (a circle with C as centre) though 0, the

⅛ ’- ►

tangent OR is the direction of the force at 0; hence by (6) aud (20), 2i the force on poq is -c?e, and acts in the direction OC; poq is therefore attracted. If the current in poq be reversed, the force will have the same numerical value, but will act in the direction CO. Hence two parallel straight conductors attract or repel each other according as the currents in them have the same or opposite directions.

⅜∙ >

Let AB (fig. 34) be an infinitely long (or very long) -> current, CD a portion of a current inclined to it, and passing very near it at 0. If the plane of the paper contain AB and CD, then at every point in OD the magnetic force is perpendicular to the plane of the paper aud towards the reader, at every point in OC perpendicular to the plane of the paper and from the reader; hence at the elements P and Q the forces acting will be in the direction of the arrows in the figure, and CD will tend to place itself parallel to AB. If both the currents be reversed, the action will be unaltered; but if the current in CD alone be reversed, it will move so that the acute angle DOB increases.

Hence it is often said that currents that meet at an angle attract each other, when both flow to or both flow from the angle, but repel when one flows to and the other flows from the angle.

These actions may be demonstrated in a great variety of ways.

Figure 35 shows an arrangement for demonstrating the attraction or repulsion of parallel currents, which is essentially that first use<l by Ampère. A is an upright consisting of a tube in good metallic connection with one of the binding screws i, and with a little cup p, containing a drop of mercury. A stout wire passes up the centre of the tube, and is insulated from it, but in metallic connec∙

tion with the screw s and the cup q. B is a light conductor, consisting of two parallelograms of wire, in which the current circulates in opposite directions, the object of which is to eliminate the magnetic action of the earth. The conductor is hung in the cups p and ç, so as to be easily movable about a vertical axis. C is a frame on which several turns of wire are wound, so that when a current is passed through, we have a number of parallel conductors, all of which act in the same way on the vertical branch uv of the movable conductor. Owing to the opposite directions of the currents in the tube and the wire inside it, there is no action on yz due to that part of the apparatus.

11 is clear, therefore, that the action of C on uv will prevail and determine the motion.

The action of straight conductors, making an angle with each other, may be shown by means of the conductor D, represented in fig. 36, which may be fitted to the stand shown in fig. 35.

In a very large class of practical cases, circular circuits play an important part. The most convenient way of dealing with these, as a rule, is to replace them by the equivalent magnets or magnetic shells. The action of a circular circuit may be represented by two layers of north and south magnetism, whose surface densities are ± i÷τ, where i is the strength of the current and τ the distance between the layers. For details concerning the calculations in a variety of cases, we refer the reader to Maxwell’s Electricity and Magnetism, vol. ii. cap. xiv.

We may calculate the force exerted, (see fig. 37) by a circular current AB on a unit north pole at its centre C, as follows. Replace the current by two discs AB and A'B', of north and south magnetism, the distance between which is τ; the surface densities are + i÷τ and -t÷τ. The first of these exerts a repulsive force 2ar*÷τ, the second an attractive force

2ιri÷τ(l - cos⅛A'C'B'); hence the resultant repulsive force is

2π√ cos ⅜ A'C'B' — τ = 2τi ÷ r, r being the radius of the disc. Hence a unit of length of the cuιτent exerts a force i÷r ’ at the distance r.

It follows therefore that the statement of our fundamental principle (p. 67) involves a unit of current strength such that unit length of the unit current, formed into an arc whose radius is the unit of length, exerts a unit of force on a unit pole placed at the centre of the arc. From this statement and the definition of a unit negative pole it follows at once that the dimension of the unit of current is [LIMIT’ 1 ].

One arrangement of circular currents has become famous from the part it plays in Ampère’s theory of magnetism. A wire wound into a cylindrical helix, such as that represented in figure 38, the ends of the wire being returned parallel to the axis of the helix, and bent into pivots, so that it can be hung upon Ampere’s stand (fig.

35), is called a solenoid. The conductor thus formed is obviously equivalent to a series of circular currente disposed in a uniform. manner perpendicular to a common axis. In the case represented in figure 38, this axis is straight; but the name solenoid is not restricted to this particular case, and what we are about to advance will apply to a solenoid whose axis is a curve of any form.

Let there be nds of the circular currents (each of area λ) in the arc ds of the axis of the solenoid. As we suppose the distribution to be uniform, n is constant. We may suppose each current to be placed at the middle of a length - of the axis, which it occupies for itself. Hence, if each circular curreut be replaced by a shell of thickness - , the surface densities of the magnetism on each of these shells will be ÷ nt, and the north magnetism of each shell will coincide with the south magnetism of the next; so that the whole action at points external to the solenoid reduces to the action of a quantity w∕λ of magnetism spread over one end of the solenoid, and a quantity - ni∖ spread over the other. The positive or north end of the solenoid is obtained, as usual, from the direction of the current, by means of the right-handed screw relation. If λ be very small, or if the system acting on, or acted upon by, the solenoid is at a distance very great compared with the dimensions of λ, then we may suppose the representative magnetism concentrated at the ends of the axis of the solenoid.

Hence the particular arrangement of electric currents, which we have called a solenoid, acts and is acted on exactly like an ideal linear magnet (whose poles coincide with the ends of its axis).

Thus the north pole of a magnet or solenoid repels the north end and attracts the south end of a solenoid; a solenoid tends to set under the action of the earth, its north end behaving like a magnetic north pole, and so on.

In a cylindrical bobbin wound to a uniform depth with silk-covered wire we have an arrangement which is equivalent to a number of solenoids all having a common axis. Each of these solenoids may be replaced by the equivalent terminal discs of positive and negative magnetism, and the external action of the whole thus calculated. The magnetic disc at each end will, of course, not be of uniform density, but if the points acted on be at a distance which is infinitely great compared with the lateral dimensions of the bobbin, we may collect the magnetism at the ends of the axis; the quantities will be

±wwιt^(α 1 + αb + δ ’), where a and b are the outer and inner radii of the shell of wire, m the number of layers in the depth, and η the number of turns per unit of length of each layer. The magnetic moment of the bob bin is therefore w 2 λi y (β s + α⅛ + ⅛ s ), where p denotes the number of turns in each layer, and mp the whole number of turns on the bobbin.

The above is a simple case· of the kind of calculation on which Weber founded his verification of Ampère’s theory. He did not, however, replace the circular currents by the equivalent magnetic distributions, but calculated directly from Ampere’s formula (18).

The instrument (electrodynamometer) which he used in his experiments was invented by himself. It consists essentially of a fixed coil and a movable coil, usually suspended in the bifilar manner, and furnished with a mirror, so that its motions about a vertical axis can be read off in the subjective manner (see art. Galvanometek) by means of a scale and telescope. Two varieties of the instrument were used by Weber. In one of these (A), the movable coil was suspended within the fixed coil; in the other (B), the movable coil was ring-shaped, and embraced the fixed coil, which, however, was so supported that it could be arranged either inside the movable coil or outside it at any distance and in any relative position with respect to it. We do not propose to go into detail respecting Weber’s experiments, but merely to indicate their general character and give some of the results. Those desiring further information will find it in §§ 1-9 of the Electrodynamische Maasbestimmungen.

Weber first showed that the electrodynamic action between two parts of a piece of apparatus traversed by the same current varies as the square of the current. Apparatus A was arranged with the plane of its fixed coil in the magnetic meridian. The movable coil was concentric with the fixed one, but its plane was perpendicular to the magnetic meridian. The current of 1, 2, or 3 Grove’s cells was sent through the fixed coil and through the suspended coil; but as the deflection with this arrangement was too great, the latter was shunted by connecting its terminals by a wire of small but known resistance. A measurement of the first power of the strength of the current was found by observing the deflection produced by the current in the fixed coil on a magnet suspended in its plane at a convenient distance north of it. After the necessary corrections were applied, the following results were obtained :— [table]

where η is the number of cells, D the electrodynamic force on the suspended coil, expressed in an arbitrary unit, M the force on the magnet, M' the force on the magnet calculated from √Dby means of a constant multiplier. The agreement between M and M' is within the limits of experimental error.

In another series of experiments Weber used the apparatus B described above. The suspended coil was arranged with its axis in the magnetic meridian, and the fixed coil set up with its axis perpendicular to the magnetic meridian. Experiments were made with the centres of the two coils coincident, and with the centres in the same horizontal plane, at distances of 300, 400, 500, and 600 millimetres, the fixed coil being, iu one set of experiments, east or west from the suspended coil; in another set, north or south. In the present series of experiments the strength of the current was measured by means of a magnet acted on, not by the fixed coil, but by another coil in circuit with it. After proper corrections, the following results were arrived at :— [table]

where d is the distance between the centres of the coils, P the couple exerted on the movable coil when the direction of that distance is perpendicular to the meridian, Q the couple when it is in the meridian. P' and Q , are the values of the same couples calculated from the theory of Ampère. The agreement here again is as near as could be expected.

Weber further showed that the deflections (v, w) of the suspended coil, calculated by means of the formulae tan V — αd~ 3 + βd~ i tan w - ⅜α√ -3 + γd- 5 in the two cases where the centres of the coils were at a considerable distance apart, agreed with observation within the limits of experimental error. Now these formula are identical with those established by Gauss for two magnets with their axes placed like the axes of the coils. This agreement therefore is an experimental proof that the coils are replaceable by magnets.

On the whole, therefore, the experiments of Weber confirm the theory of Ampere, as far as experiment can test it. They form, therefore, a sufficient basis for the proposition on which we founded our theory; for this proposition leads to the same result for closed circuits as the theory of Ampère.

The action of any current on a magnetic pole, and hence on any magnet, may be calculated either by replacing the circuit by an equivalent shell or by means of formula (19). We have already found this action in the particular case of an infinitely long straight current. This result was originally found experimentally by Biot and Savart, and Laplace showed that it followed from their result that the force exerted by an element of the current varies inversely as the square of the distance. The fact that a circular current acts on a magnetic pole at its centre in the same way as a zig-zag current which is everywhere very nearly coincident with it, leads, when properly interpreted, to the conclusion that the force varies as sind In this way formula (16) was originally arrived at, independently of Ampère’s theory.

A great variety of instances might be given of the action of a magnet on a current. The earth, for instance, acts on a circular current, hung up on Ampère’s stand: the current, being movable about a vertical axis, will turn until the maximum number of the earth’s lines of magnetic force pass through it—ι.e., it will set with its plane perpendicular to the magnetic meridian, in such a way that the current, looked at from the north side, goes round in the opposite direction to the hands of a watch.

A very simple way of showing the action between magnets and currents was devised by De la Rive. A small plate of copper and a small plate of zinc are connected together by a wire passing through a cork and making a circuit of several turns; the cork is placed in a vessel containing dilute sulphuric acid, and floats on the surface, carrying the little circuit about with it. Such a circuit will set under the earth’s action, and may be chased and turned about, &c., by a magnet. After what has been already said, however, such experiments offer no new point of interest.

Electromagnetic Potations. — It is obvious that no invariable system of electric currents can produce continuous rotation of a magnetized body. For, suppose an elementary magnet, whose action may be represented by two poles of strengths ÷ m, to describe any path and to return exactly to its former position; either it has or has not embraced the circuit in its path; if it has not, no work has been done on either pole; if it has embraced the circuit n times, an amount of work 4nwτri has been done on the north pole, and an amount - 4τm7ri on the south; on the whole, therefore, no work has been done on the magnet. As any magnetized body may be conceived to be made up of such elementary magnets, it is obvious that it is impossible for such a body to rotate continuously, doing work against friction, &c.

The same is obviously true if we replace the magnet by an invariable system of electric currents.

If, however, part of the electric circuit is movable with respect to the rest, and communicates therewith by means of sliding contacts or the like, continuous rotation of part of the circuit may occur. Again, if by any artifice the magnet can be transferred every revolution from one side of the current to the other, continuous rotation of the magnet may result. Lastly, if the direction of the currentin some part of the apparatus be always reversed at a certain stage of the revolution, continuous motion may ensue.

Rotations of the first and second class were first discovered by Faraday, and the ground principle of most of the pieces of apparatus used in demonstrating them is that originally used by him.

One of the simplest cases is the rotation under the action of the vertical component of the earth’s magnetic force. Let ABC (fig. 39) be a horizontal circular conductor, OP a conductor pivoted at O, having sliding βontact at Γ with A BC. Let a current i enter ABC at A, and leave it at P, flowing through PO to O and thence to the battery again. The magnetic force at any element dr of OP is perpendicular to OP and to the plane of ABC, hence the electromagnetic force on the element will be in the plane of ABC, in the direction of the arrow p, and will be equal to iRdr (R = vertical component of earth’s force). Hence the moment about O of the forces acting on OP is fiRrdr, i.e. ⅜OP 2 Rr, which is independent cf the position of OP. OP will therefore rotate about O, with an angular velocity which will go on increasing until the work lost by friction, &c., during each revolution is equal to πOP 2 Rr.

Ampère has given a general theory of the rotation of a circuit under >—*∙

the action of a magnet. Tuet AB (fig. 40) be any circuit, which we may suppose connected with the axis of the magnet, but free to rotate about it. We suppose the magnet replaced by quantities ± τη of magnetism at its poles. Take the axis of the magnet for axis of z, and the other axes as in the figure, 0 being the centre of the magnet, and let ON = OS = c. Let PQ be any arc ds of AB, and let the coordinates of P be x, y, z; then if l, m, n be the direction cosines of NP, and NP-=D, we have Dl=x, Orn = y, Dn = z - c; also the direction cosines of Pp, which is perpendicular to NP and PQ, and is the direction of the

∕ dy dz∖

force exerted by the pole N on P, are (n c j s ~ τn )÷sinQPK, &c.

Hence by formula (6) the components of the force acting on PQ are m f dy dz∖, , -D∖ n cls ~ m ds) ldβ > fic -

Hence, if K denote the moment of these forces about OZ, we have from the north pole alone

. fds∖ ∕,dz dx∖ f dy dz∖ ) κ - m y iPi V*- n a) χ -("⅛- n ∙<⅛,)≈' j∙

If we substitute the values of l, m, n this reduces to

K ≈mij'dsf^ =rnijdn.

If therefore β i , α j , β 2 , a , i denote the angles BNZ, ANZ, BSZ, ASZ, we have, adding the results from both poles,

K=τnι (cos 3ι - cos αj - cos β 3 + cos α 2 ) . . (21).

_ It follows from this remarkable formula that the couple K tending to turn a part AB of an electric circuit about the axis of a magnet depends merely on the position of the ends A and B.

In particular, if A coincide with B, t.β. if AB form a closed circuit, or if A and B both lie on parts of the axis not included between N and S, the couple will be nil, and there will be no rotation.

The application of this formula to cases where there are sliding contacts at A and B not lying on the axis presents no difficulty; we leave it· to the reader.

Several of these rotations may be exhibited by means of the apparatus represented in figure 41. A BC is a horizontal coil of wire terminating at the binding screws a,b. FG is a ring-shaped trough of mercury for the sliding contacts. A wire connects the mercury with the binding screw d. DE is an upright support screwed into a metal base D in connection with the binding screw c, and terminating above in a mercury cup E. When required, DE can be replaced by the shorter supports D'E'and DE". H LK is a support for a screw L, which carries an adjustible centre.

1. Poise in the cup E the wire stirrup MN, so that the ends just dip in the mercury trough. Then, if a strong current be sent from c to d, MN will rotate (iu northern latitudes) in a direction opposite to the hands of a watch.

2. If we fix a vertical magnet n ’ "d" to DE by means of a clip at Y, then the rotation will take place with a weaker current in the same direction as before, if the north pole of the magnet be upwards (as shown in figure), but in the opposite direction if the magnet be reversed.

3. Reversing the current alone in either of the last two cases causes the direction of rotation to be reversed.

4. The magnet may be removed and a current sent from a to b round ABC in the direction opposite to the hands of a watch. The result is the same as for the magnet with its north pole upwards. If the current in ABC is reversed, the rotation is reversed; and so on.

5. The support D'E' with the two magnets ns, nd may be screwed into D instead of DE, the wire P now dij>ping into the mercury. If the current be sent from c to d, the vertical current in D'E' will act on s and s', and cause the magnet to rotate in the direction of the hands of a watch. This rotation is reversed if the current go from d to c.

6. We may consider any magnet of finite size as made up of a series of magnets like ns and n ’ d arranged about an axis. Hence, if we replace D'E' and the magnets D"E" by the single magnet supported by means of the pivot L”, there will still be rotation.

Figure 42 represents a very elegant piece of apparatus devised by Faraday, to show the rotation at once of a magnet and of a movableconductor. The rotating pieces are the magnet sn, which is tied to the copper peg at the bottom of G by means of a piece of string, and swims round the vertical current buoyed up by the mercury in G, and the wire DE, which is hinged to D by a thin flexible wire, and swims round the pole of the vertical magnet n ’ d.

Another apparatus invented by Barlow, and known by the name of Barlow’s wheel, is represented in figure 43. A current is caused to pass from the mercury trough C along the radius of the disc A through the field of magnetic force due to tho horse-shoe magnet NO. The result is that the wheel rotates in the direction indicated by the arrow.

Fluid conductors may also be caused to rotate under the action of a magnet. We mentioned 4 in our historical sketch the experiment by which Davy demonstrated this rotation in the case of mercury. A variety of such experiments have been since devised. The following is a simple one. Fill a small cylindrical copper vessel with dilute sulphuric acid and set it upon the north pole of a powerful electromagnet. If a thick zinc wire be connected by a piece of copper wire to the copper vessel, and then immersed in the acid so as to be in the axis of the vessel, a current is set up in the liquid which flows radially from the zinc to the copper across the lines of force.* The liquid therefore rotates in the direction of the hands of a watch.

Magnetic Action on the Electric Discharge in Gases.—A large number of very interesting results have been obtained concerning the behaviour of the electric discharge in a field of magnetic force. We can only make a brief allusion to the matter here. The key to the phenomena lies in the remark that the electric discharge in vacuum tubes may be regarded as an electric current in a very flexible elastic conductor. It is clear that such a conductor would be an equilibrium if it lay in a line of magnetic force passing through both its fixed ends. Again, if the flexible conductor be constrained to remain on a given surface, it will not be in equilibrium until it has so arranged itself that the resultant electromagnetic force at each point is perpendicular to the surface. At each point, therefore, the magnetic force must be tangential to the surface.

A perfectly flexible but inextensible conductor, two points of which are fixed, will take such a form that the electromagnetic force at each point is balanced by the tension. Le Roux fastened a thin platinum wire to two stout copper terminals, and caused it to glow by passing a current through it. When the terminals were placed equatorially between the flat poles of an electromagnet, the wire bent into the form of a circular arc joining the terminals. When the terminals were placed axially, it assumed a helical form. (See also Spottiswoode and Stokes, Proc. H S., 1875.)

The behaviour of the light emanating from the positive pole may be explained in general as lying between the two cases which we have just discussed. One of the most remarkable of these phenomena is the rotation of the discharge discovered by Walker, and much experimented on by De la Rive. This may be exhibited by means of the apparatus shown in fig. 44, consisting essentially of an exhausted vessel, one of the electrodes in which is ring-shaped; a bar of soft iron, covered with some insulating material, is passed through the ring and fixed to the stand. When this apparatus is placed on the pole of a powerful magnet, the discharge rotates as a wire hinged to the upper electrode would do.

Owing to the distinct character of the negative light, the action of the magnet on it is different from that on the positive light. Phicker found that the general character of the phenomena may be thus described :— The negative light is bounded by magnetic curves that issue from the electrode and cut the walls of the tube.

The two diagrams in fig. 45 will convey an idea of the appearance of the phenomenon. Although much tempted to follow the subject further, we must be content to refer the reader to the interesting papers of Phicker and Hittorf. An excellent summary will be found in Wiedemann.

Ampere’s Method.—Before quitting the subject of electromagnetism, it will be useful, for the sake of com- ’ parison, to give a brief sketch of the method of Ampère, or rather of that modification of the original method now commonly found in Continental books, which was suggested by Ampère himself, in a note to the Théorie des Phénomènes Electrodynamiques. Ampère starts with the idea that the electrodynamic action of two circuits is the sum of the actions at a distance between every pair of their elements. He supposes, as the simplest and most natural assumption, that the force between two elements is in the line joining them. Besides this assumption, his theory rests on four experiments. The first of these shows that, when a wire is doubled on itself, the electrodynamic action of any current in it is nil. The second experiment shows that this is also true, even if one of the halves of the wire be bent or twisted in any way, so as never to be far removed from the other. The third experiment proves that the action of any closed circuit on an element of another circuit is perpendicular to the element. In the fourth experiment it is shown that the force between two conductors remains the same when all the lines in the system are increased in the same ratio, the currents remaining the same. From the assumption, together with the first experiment, it follows that the force between two elements is proportional to the product of the lengths of the elements, multiplied by the product of the strengths of the currents and by some function of the mutual distance and of the angles which determine their relative position. Hence it may be shown, from the fourth experiment, that the force between the elements must vary inversely as the square of the distance between them. The second experiment shows that we may replace any element of a circuit by the projections of the element on three rectangular axes.

From these results it is found that the force between ds and dσ must be

∣^ ’ ' ff (cos e - k cos θ cos θ'}.

The constant k is then determined from the result of the third experiment; and it is found that k must be equal to ⅜. The formula is thus completely determined, with the exception of A, which depends on the unit of current which is chosen. The action of a closed circuit on an element is then calculated, and a vector found, which Ampère calls the “directrix,” from which this action can be found in exactly the same way as we derived this same action from the magnetic induction. The theory is then applied to small plane circuits, solenoids, and so on.

As was remarked in the historical sketch, a variety of other elementary laws may be substituted for that of Ampère, all of which lead to the same result for closed circuits.

Maxwell has presented Ampère’s theory in a more general form, in which the assumption about the direction of the elementary action is not made. Neglecting couples, he finds for the most general form of the components of the force exerted by dσ on ds, "-di( * ⅛ 23 ⅛Λ' 'ω, ,,fe '

in the direction of D, « 1 00 ×

c , dQ..,,, α' d ⅛' 1 J l’’

and S= --^acdsdσ ’ S = ^ s ιιdsdσ ⅛

in the direction of ds and dσ respectively. ∕

In these expressions Q is a function to be determined only by further assumption. Q = constant gives Ampere’s formula; Q = - gives the formula of Grassmann, and so on. We may in fact construct an infinite variety of different elementary formulae. The reader interested in this subject may consult Wiedemann, Bd. ii. §§ 26, 27, 45-54, <kc., and Tait, Proc. R.SE., 1873.

In our account of the magnetic action of electric currents no mention has been made of the effect of the proximity of soft iron. Under the magnetic action of the electric circuit soft iron is magnetized inductively. The distribution of the lines of force is in general greatly affected thereby. The general feature of the phenomenon is a concentration of the lines upon the iron. By the proper use of this effect electromagnetic forces of great power may be developed. It is not easy to give a mathematically accurate account of the action, owing to our ignorance of the exact law of magnetic induction in powerfully paramagnetic bodies. The discussion of this subject, however, belongs to Magnetism (which see).

The Induction of Electric Currents.

A brief account has already been given (see Historical Sketch, p. 11) of Faraday’s discovery of the induction of electric currents. The results he arrived at may be summed up as follows.

Let there be two linear circuits, ABKE (the primary) and CDG (the secondary), two portions of which, AB and CD, are parallel, and near each other.

I. When a current is started in A B, a transient current flows through CD in the opposite direction to the current in AB; when the current in AB is steady, no current in CD can be detected; when the current in AB is stopped, a transient current flows through CD in the same direction as the current in AB. These currents in CD are said to be induced, and may be called inverse and direct currents respectively, the reference being to the direction of the primary. Both inverse and direct currents last for a very short time, and the quantity of electricity which passes in each of them is the same.

II. If the circuit AB, in which a steady current is flowing, be caused to approach CD, an inverse current is thereby induced in CD; when the circuit AB, under similar circumstances, recedes from CD, a direct current is induced in CD. We have already mentioned that when AB is at rest, and the current in it does not vary, there is no current in CD. AB has been supposed to approach and recede from CD, but the same statement applies when CD approaches and recedes from AB.

III. When a magnet is magnetized or demagnetized in the neighbourhood of a circuit, or approaches or recedes from the circuit, the effect is the same as if an equivalent current approached or receded from the circuit. For example, imagine a small circular circuit placed horizontally, and a vertical bar magnet lowered in the axis of the circuit with its north pole pointing down upon the circuit, the magnet may be replaced by a series of coaxial circular currents (see above, p. 71), and the motion will induce a current passing round the circuit against the hands of a watch.

Faraday showed how the direction of the induced current can be predicted when the variation of the magnetic field or the motion of the conductor in it is known, and he gave, in his own manner, indications how the magnitude of the current could be inferred.

Maxwell has thrown the law of Faraday into the following form :—“The total electromotive force acting round a circuit at any instant is measured by the rate of decrease of the number of lines of magnetic force which pass through it.”

Or, integrating with respect to the time :—“The time integral of the total electromotive force acting round any circuit, together with the number of lines of magnetic force which pass through the circuit, is a constant quantity.”

For “number of lines of force” may of course be substituted the equivalent expressions, “induction through the circuit,” or “surface integral of maguetic induction,” taken over any surface bounded by the circuit.

Some care must be taken in determining the positive direction round the circuit. The following is a correct process :—Assume one direction (D, fig. 46) through the circuit as positive, then the positive direction round (R) is determined by the right-handed screw relation; if the number of lines of force reckoned positive in direction D is decreasing, then the electromotive force is in direction R; if that number is increasing, the electromotive force is in the opposite direction.

This will be clearer if we consider the following simple example. Let ABCD (fig. 47) be a horizontal rectangular circuit (AB next the reader). In a northern latitude, the vertical component Z of the earth’s magnetic force is downwards; if, therefore, the positive direction through the circuit be taken downwards, the positive direction round is ADCB, and the number of lines of force through it is Z.AB.BC. If

BO slide on DC and AB parallel to itself through a small distance BB' in time τ, Z.AB.BC increases by Z.CB.BB'; hence the electromotive force is Z.BC.BB'÷τ, and acts in the direction ABCD. If V be the velocity of BC, we may write for the electromotive force Z. BC.v. That is, the electromotive force at any instant is proportional to the velocity.

The law of Faraday leads to a complete determination of the induced current in all cases. We may regard it as resting on the experiments of Faraday, and of those who followed out his results.

Another view of the matter of great importance was enunciated independently and about the same time by Helmholtz and Sir William Thomson.

Let a circuit carrying a current! move in an invariable magnetic field, so that the number of lines of magnetic force passing through it is increased by <ZX, then the work done by the electromagnetic forces on the circuit is by Ampere’s theory f<fN; also, if li be the resistance of the circuit, 'Ri-dt is the heat generated in time dt. Now if E be the electromotive force of the battery which maintains the current i, the whole energy supplied is Vidl; hence we must have f<7N + Rι 2 dt-Vidt

, . ∕ τ , dN∖ 1 a ∏dt-( E dt ) κ .... (23).

dN

Hence there is an electromotive force in the moving circuit. dN

Now is the rate of increase of the number of lines of force pass ing through the circuit.

We have therefore deduced the law stated above from Ampère’s theory .and the principle of the conservation of energy; at least we have doue so for the case of induction by permanent magnets, and the same reasoning will also apply to the case where the alteration of the magnetic field, owing to the induced current in the primary circuit, is so small that it may be neglected.

We have now the means of stating in a convenient form the electromagnetic unit of electromotive force. It is the electromotive force of induction in a circuit the number of lines of magnetic force through which is increasing at the rate of one per second.

In the case where the field is due to a current t', we have by formulae (4) and (14) of last division

N=i'M (24), where M now stands for fj^dsdσ extended all over the two circuits. M, which depends merely on the configuration and relative position of the two circuits, is called the coefficient of mutual induction .

An application of the principle of the conservation of energy of great importance wa≡ made by Sir William Thomson to the case of two electric circuits of any form, in which the currents are kept constant.

Let two such circuits, the currents in which are tι', be displaced so that the coefficient of mutual induction M increases by dM. Let us suppose that the currents i and »' are maintained by two constant batteries of electromotive forces E and E', aud that the motion takes place so slowly that the currents may be regarded as constant throughout. If R and R' be the resistances of the circuits, H<fi the mechanical equivalent of the whole heat generated, and KrfZ the whole expenditure of chemical energy in the batteries in time dt,

H = Ri* + R'ι* 2 , and K = Ei + E't', K-H-t(E-Rι∙) + i(E-RT).

Now, applying (23),

Ri=E -1' ⅛- , and R'i' - E' - Ä; at at whence K — H = 2U' , or, as we may write it,

(K-H)<fc=2tt'dM ... . (25).

Now ii'diA. is the work done by the electromagnetic forces during the displacement which we may suppose spent in lifting a weight.

Hence, when two electric currents are allowed slowly to approach each other, being kept constant and doing work the while, over and above the work which is spent in generating heat in the conductors, an amount of energy is drawn from the batteries equivalent to twice the work done by the electromagnetic forces.

There remains therefore an amount of work as yet unaccounted for. What becomes of it Î The answer is, that the energy, or, as Sir W. Thomson calls it, the “mechanical value,” of the current is increased. But how increased? When a material system (and we may consider the tλvo circuits, the batteries, the lifted weight, <fcc., as such) is left to itself, it moves so that its potential energy decreases. In this case, therefore, there must have been an increase of kinetic energy somewhere. This energy may be called the electrokinetic energy of the system; according to Maxwell’s theory, this kinetic energy has its seat in the medium surrounding the wire. The energy thus stored up is accounted for in the increased development of heat, ιtc., when the two currents are broken in succession.

Returning now to our general law of induction, let us w rite down in the most general form the equations which determine the course of the currents in two circuits (A, B), in which the form and relative positions of the circuits, as well as the current strengths, are variable. The number of linee of force which pass through a circuit depends partly on neighbouring circuits, partly on the circuit itself. Retaining the notation used above, we may, in the case of two circuits, write the first part Mi', and the second part Li; where L is a double integral of the same form as M, only both elements ds and dσ now belong to the same circuit. We have, therefore, for the whole number of lines of force passing through the circuit A, Mi' + Li. Similarly we have for B, Mi + Nz". We have therefore by our general law,

E - (Mi' + Li) = Ri )

d i · · · < 26) ·

E ' - ( Mi + N i')=R'i' )

These are the general equations for the induction of two circuits. The electromotive force of induction in A can be divided into two parts: one of these, viz., i ^(Mι') is due to the circuit B, the other (Li) is due to the circuit A itself, and is called the electromotive force of self-induction. L is called the coefficient of self-induction for A. Similarly ~^(Nι') is the electromotive force, and N the coefficient force of self-induction for B.

If we have only one circuit then M = 0, and the equation for the course of the current is E " ; here there is only self-induction.

F. E. Neumann, to whom belongs the honour of first stating with mathematical accuracy the laws of induction, adopted a foundation for his theory very different from the one chosen above. His method was based on the law of Lenz, enunciated very soon after the great discovery of Faraday, which lays down that, in all cases of induction by the motion of magnets or currents, the induced current has a direction such that its electromagnetic action on the inducing system tends to oppose the motion producing it.

Besides its historical importance, this law affords a very convenient guide in many practical applications of the theory of induction. The reader will find no difficulty in verifying it on the elementary cases given at the beginning of this division. It can be deduced at once from our general law. Consider any circuit in which a current i is flowing, and let the direction of the current be the positive direction round the circuit. Suppose the circuit to move so that the number of lines of force passing through it increases, this is the way the circuit would tend to move under the electromagnetic forces when traversed by a current i; but the electromotive force of induction is in the negative direction round the circuit by the general law, and would therefore produce a current opposite in direction to i. The electromagnetic action on this current would be opposite to that on i, that is, would tend to hinder the displacement. It is a curious fact that a law exactly like this had been announced shortly before Lenz by Ritchie, only with the direction of the action reversed in every case.

The results of Neumann are identical with those given above. The double integral M, which is here called the coefficient of mutual induction of two circuits, Neumann calls the mutual potential of the two circuits, and what has been c filed above the coefficient of self-induction of a circuit he calls the potential of the circuit on itself. Accounts of his theory will be found in Wiedemann’s Galvanismus, and in most Continental works on electricity.

Experimental Verification of the Laws of Mutual Induction.. —It will be observed that, in the law of induction for linear circuits, no statement is made respecting the material or thickness of the circuit in which the electromotive force of induction acts, or of the non-conducting medium across which induction takes place.

Faraday showed that the material of the circuit has no effect. He found, for instance, that when two wires of different metals were joined and twisted up together, as in fig. 48, so as to be insulated from each other, no induced current could be obtained by passing the arrangement between the poles of a powerful magnet. The same result was obtained when one of the branches of the circuit was an electrolyte. Lenz connected two spirals of wire in circuit with each other, and placed first one then the other, on the soft iron keeper of a horse-shoe magnet; so long as the number of turnson each spiral was the same, the induced current was the same, no matter what the material or thickness of the wire in each spiral. Since in this case the whole resistance of the circuit was always the same, the electromotive force of induction must have been the same.

We conclude, therefore, that the electromotive force of induction is independent of the material, and also of the thickness of the wire, so long as the latter is so small that we may consider the wire as a linear circuit.

Lenz made quantitative determinations of the induced current by means of the above arrangement.

The soft iron keeper, with a coil of n windings, was rapidly detached from the magnet, and the first swing a of a galvanometer in circuit with the coil was measured. The quantity of electricity which passes in the induced current is measured by sin ⅜α, provided the whole duration of the current is small compared with the time of oscillation of the galvanometer needle (see art. Galvanometer). Again, when the keeper is attached to the magnet, very nearly all the lines of magnetic induction pass through the keeper; hence the number of lines of induction which pass through the coil is very nearly proportional to the number of windings, and therefore, if the resistance of the circuit be kept the same, the whole amount of electricity which passes will be proportional to n. In the actual experiment the wire was wound and unwound from the keeper, so that the whole resistance did remain the same. The following is a set of Lenz’s results :— [table]

The value of sin∣α÷n is very nearly constant. It increases a little as the number of windings increases, as ought to be the case, for, although most of the lines of induction pass through the keeper, yet all do not, and a few more are included when the number of turns 13 increased.

Faraday made special investigations in search of the effect of the medium across which induction is exerted. He found that no effect on the integral current was produced by inserting shellac, sulphur, copper, &c. between the primary and secondary coils. The insertion of iron or any strongly magnetic body, of course, produces an effect, because the distribution of the lines of magnetic force is thereby altered, and therefore, by our general law, the electromotive force of induction will be correspondingly affected. We conclude, therefore, that the electromotive force of induction is independent of the medium across which it is exerted.

It must be remarked, however, that in the case of conducting media, the statement is subject to a certain limitation, the nature of which follows from the law of induction itself. For there will be induced currents in the intervening medium if it be a conductor, and these currents will disturb the lines of force while they continue to flow. These currents are transient, however, so that their integral effect on the number of lines of force passing through the secondary is zero. It is obvious, therefore, that, if we replace “electromotive force” by “time integral of electrυ-tive force extended over the whole time that the induction currents last,” the statement will still be true. The only effect, therefore, of interposed conducting media is on the time which the induced currents take to rise and fall.

Weber applied his electrodynamometer to test the laws of induction.

The suspended coil was caused to oscillate when there was no current either in it or in the fixed coil, and the logarithmic decre ment of its oscillations carefully determined. This decrement, due to the friction of the air, &c., was found to be constant for different lengths of the arc of oscillation. The terminals of the suspended coil were next connected so that it formed a closed circuit, and a constant current was sent through the fixed coil. Induction currents were now generated in the suspended coil, whose electrodynamic action constantly opposed its motion. It was found that the logarithmic decrement was still constant, but greater than before. Weber therefore concluded that the induced current at each instant was proportional to the velocity of the coil. Since the resistance does not vary, this is in accordance with the general law.

Weber further showed that the induced current is the same whether it is produced by a current in the fixed coil or by a magnet, which exercises the same electromagnetic action as that current on the suspended coil, when the latter is traversed by a current of unit strength.

The electrodynamometer may also be used to demonstrate the equality of the whole amounts of electricity which pass in the direct and inverse currents. If the induced currents from a secondary coil whose primary is being “made and broken” be passed through both coils of the instrument, there will be a deflection, since the action depends on the square of the current; but if the induced current be sent through the suspended coil alone, and a constant current be sent through the fixed coil, there will be no deflection, which shows that the quantities of electricity passing in the alternate currents of the secondary coil are equal and of opposite sign.

Felici (1852 and 1859) made an extended series of experiments on the laws of induction. He used null methods, and his experiments bear a resemblance in some respects to the electrodynamical experiments of Ampère. Maxwell has given a summary of Felici’s results.

It is found, for instance, that the electromotive force of induction of a circuit A on another B is independent of the material or section of the conductors, that it is proportional to the current in A and to the number of windings in B. The induction of A on B is the same as that of B on A, wlιen the inducing current i is the same in both cases. Any portion of A or B may be replaced by a zig-zag portion, which nowhere deviates far from it. In pairs of circuits geometrically similar, the electromotive force of induction is proportional to the linear dimensions, and so on.

if B be so situated with respect to A that starting or stopping a current in A produces no induced current in B, B is said to be conjugate to A. There are an infinite number of such conjugate positions of B; and Felici shows that, if B be moved from one of these P 1 into another P s very quickly, no effect is produced on the galvanometer. If B be moved from P l to any position P (not a conjugate ∣>osition), the effect on the galvanometer is the same as if the current i were suddenly started in A, B being in the position P.

All these results are direct consequences of our general law, and indeed might be used as a foundation for it.

In his later researches on electromagnetic induction (series xxvii. and xxix.), Faraday develops in considerable detail his ideas on the connection between the lines of magnetic force and the induced current, and gives increased precision to the experimental methods that flow therefrom. He points out the great value of methods, such as the use of iron filings, for exhibiting in a visible form the course of the lines of magnetic force. He also insists on the great use of a small moving circuit, which can be used to explore the magnetic field under circumstances which render the application of other methods impossible.

The direction of a line of force may be determined in various ways by means of the moving conductor. Maxwell gives four such ways :—(1) if a conductor be moved along a line of force parallel to itself, it will experience no electromotive force; (2) if a conductor carrying a current be free to move along a line of force, it will show no tendency to do so; (3) if a linear conductor coincide with a line of force and be moved parallel to itself in any direction, it will experience no electromotive force in the direction of its length; (4) if a linear conductor carrying an electric current coincide in direction with a line of magnetic induction, it will not experience any mechanical force.

In these researches Faraday treats at considerable length a case of the induction of electric curreuts, to which Continental writer's have given the somewhat mysterious name of “unipolar induction.” It belongs to a class of cases on which they have rightly dwelt as being in a sense the reverse of the electromagnetic rotations. The following theory of the phenomenon will make this clearer :—

Referring back to figure 40, let AB be part of a conducting circuit arranged as there described, and let it be caused to move in the direction 1⅛. Then if E be the electromotive force in the circuit in the direction AB, N the number of lines of force passing through the circuit, <ρ the angle through which AB moves (from X to Y) about OZ, we have, by our general law,

_dN = dφ di dφ di idN

Now, by Ampère’s theory, K = > heuce (p. 73)

p = _ κ ⅛> i di

— - in (cos 3ι - cos α, - cos β. 1 + cos c⅛) ~y . (27).

Hence, if the conductor AB be caused to move with given angular velocity about the magnet SN, in that direction which it would take under the action of the magnet if it carried a current i, then there will be an electromotive force of induction along the circuit of which AB forms part, whose direction is opposite to that of i, and whose magnitude is found by dividing the couple acting on AB (when traversed by i) by i, and multiplying it by the given angular velocity. This result is a beautiful instance of the law of Lenz.

A great variety of experimental arrangements may be imagined to realize the case thus described. Every apparatus devised to produce an electromagnetic rotation may be used to illustrate it.

The following case may be taken as typical. SN (fig. 49) is a bar magnet whose action may be represented by two poles, N aud S. At the middle point of its axis is fixed a disc BA, against which presses the terminal of a wire CA in metallic connection with the axis through the pivot at S. If CA be caused to rotate in the direction of the arrow p, the disc standing still, there will be an induced current in CABC in the direction of the arrow q. If CA and the disc revolve together, there will be no current. If CA stand still, and the disc rotate in the direction of the arrow, there will be a current in the opposite direction; for this is clearly the same as if the disc stood still, and CA rotated in the opposite direction. The electromotive force in each case is independent of the form of CA, and is given by 2jh(1 -cosα)ω, where m is the strength of the pole N, a the angle ANB, and < t > the angular velocity.

It is well to remind the reader that the lines of force are closed curves, every one of which passes up the axis of the magnet from 8 to N, and back through the outside medium to S. If this be forgotten, and an attempt be made to determine the electromotive force of induction by considering the motion of the disc, an error will easily be made. If we take the simpler course above, and consider the motion of the conductor, there is then no danger of mistake.

In most of the experiments we have hitherto been describing, the object has been to obtain indications of the direction of the currents of induction, or to measure the electromotive force of induction under definite circumstances; if, however, we desire to exhibit the effects of induction in a striking manner, in order to convey belief to the spectator, or to serve some practical purpose, recourse is had to a different kind of apparatus. We may wind our primary and secondary coils on bobbins, and insert the former w ithin the latter, so as to get the greatest possible number of turns of wire into proximity. The number of turns on the primary is usually made small, in order that the current in it may not be weakened by a large resistance, and that its coefficient of self-induction (see below) may be small. Mention has already been made of the effect of soft iron in increasing the number of lines of force that pass through a circuit. It is easy to see that it will produce a corresponding effect in strengthening induction. The precise amount of it is very hard to calculate, owing to the irregularities in the magnetization and demagnetization that arise from residual magnetism. The question belongs, however, to magnetism. The effect can be demonstrated practically by observing the alteration in the inductive action produced by inserting a bundle of iron wires into our primary coil.

The physiological effects of induced currents are very striking; indeed, the nerve and muscle preparation of the physiologist affords a very delicate method for detecting them. If the human body form part of the circuit of the secondary coil of such an induction apparatus as we have just indicated, and the primary current be stopped and started in rapid succession, say by stripping one terminal of the circuit on a toothed wheel attached to the other, a sensation is experienced which, with a moderately powerful apparatus furnished with a core, is so painful and peculiar that the patient is not likely to forget either it or its cause. The tetanic muscular contractions produced in this way have formed the subject of much physiological investigation, of which an account will be found in the proper place (see article Physiology).

The fiat spirals of Henry, formed of flat bands of copper insulated from each other with silk ribbon, are also very convenient for demonstrating the existence of induced currents.

The most powerful inductive apparatus for furnishing large quantities of electricity are the various magnetoelectric machines which have now been brought to great perfection (see Historical Sketch).

By means of these and similar appliances, all the effects of the electric current and the electric discharge may be shown in the greatest perfection.

Induction by Discharge of Statical Electricity.—The phenomena of induction can be exhibited with the transient current of electricity in the discharge of a Leyden jar or other accumulator of statical electricity. There is a difficulty in exhibiting the effect, owing to the great differences of potential between different parts of the circuit, which render the application of a coil of silk-covered wire useless. A common way of getting over the difficulty consists in cutting two spiral grooves in two flat ebonite discs. Wires are embedded in these, and they are then put together with a thin plate of glass between, so that the spirals are opposite each other. When a jar is discharged through one spiral, an induction current passes in the other, and may be indicated by a galvanometer, or, better still, by a frog preparation. The induced current is, however, in general a complicated phenomenon, owing to the oscillatory nature of the discharge (see above, p. 65).

It would lead us too far to go into these and kindred subjects: the reader who desires to pursue the matter will find excellent accounts in Mascart, t. ii. §§ 611-825, and Riess, Bd. ii. §§ 780-906. Particularly interesting are the researches of Verdet, an account of which will be found in his works, along with many indications of ∖Jιat others have done in the same field.

Induced Currents of Higher Orders.—Induced currents may in their turn induce other currents, and these again others, and so on. This can be brought about by forming part of the secondary circuit of one inductive apparatus into the primary of the next, and so on. As may be supposed, the successive induced currents diminish very rapidly in strength, and require special means for their detection. But the phenomenon also goes on increasing in complicacy. Suppose we start the current in the first primary, there is a single inverse current of the “first order” which rises and then falls; there will, therefore, be two currents of the “second order ”—first a direct, then an inverse; each of these rising and falling causes two currents of the third order, and so on in geometric progression. These currents have been detected in certain cases by means of their physiological action and their magnetizing powers. The latter effects present some points of interest in connection with magnetism, but we cannot spare space for the matter here.

Self-Induction.—The existence of self-induction has been deduced as a theoretical consequence of the general law of induction. It was not so discovered, however. It was first arrived at by Faraday from experimental considerations. The observation from which he started was the following fact communicated to him by Mr Jenkin, who had shortly before discovered it :—Although it is impossible with a short circuit of wire and a single battery cell to obtain a shock by making and breaking contact, yet a very powerful shock is obtained if the coil of an electromagnet be included in the circuit. This may be shown thus :—Let ZC (fig. 50) be a battery of a single cell, CABDEF a circuit with a cross branch BF, in which at G the human body, &c., may be inserted. Contacts can be made and broken at A, very rapidly if need be, by means of a toothed wheel. When BDEF consists of a short single wire, nothing particular is felt at G, but when the coil of an electromagnet is inserted in DE, the patient at G experiences a series of powerful shocks comparable to that obtained from the secondary coil of an inductive apparatus in the manner already described.

If the cross circuit be done away with, a powerful spark is obtained at A on breaking contact, but none on making. This spark is particularly bright if a mercury contact be used, owing to the combustion of the mercury. If, however, the electromagnet be removed from DE, and a short wire substituted, the spark becomes quite insignificant, although the whole circuit may be now very hot, owing to the increased current. Faraday found that the same effect, only smaller, was produced when a simple helix without a core was substituted for the electromagnet; and a similar effect, only still smaller, was obtained when a very long straight wire was used. Faraday soon recognized that these effects are consequences of the laws at which he had arrived in his first series of researches on induction. When the current is rising in a circuit, the number of lines of magnetic force passing through it is on the increase, hence an electromotive force is generated which opposes that of the battery, and causes the current to rise slowly; again, when the current begins to decrease the number of lines of force begins to decrease, and an electromotive force of induction is called forth which tends to prolong the current. We have, therefore, a weakening of the electromotive force at starting and an exaltation at stopping, which accounts for the absence of the spark or shock at make, and the presence of one or other at break. Such inductive effects are obviously heightened when the cur rent is wound into a spiral form; if, however, the spiral were wound double, and the current sent through the two wires in opposite directions, the inductive effects would annul each other, and, in fact, with this arrangement the spark and shock are extremely small.

Faraday demonstrated the existence of these electromotive forces by means of the currents which they produce in derived circuits, when the battery contact is broken or made.

He used the arrangement given in figure 50. A galvanometer was inserted at G, and the needle stopped by pins properly placed from deviating as urged by the branch of the battery current from B to F, but left free to move in the opposite direction. It was found that the needle deviated sharply when contact was broken at A, in a direction indicating a current from F to B. Again the contact was made, and the needle stopped at the deviation due to the current from B to F, so that it could not return to zero. The contact was then broken and made again, and it was found that at the make the needle tended to go beyond the position due to the steady current in BF. Faraday also arranged a platinum wire at G, so that it did not glow under the steady current in BF, but immediately ignited when the contact at A was broken. Chemical action was produced in a similar manner. In fact we may, by taking advantage of the self-induction, cause a single cell to produce decomposition of water and evolution of gas, which it could not do alone consistently with the conservation of energy. This may be managed by inserting at A (fig. 50), instead of the contact breaker, the coil of an electromagnet, and placing the decomposing cell in DE. Let contact be made and broken at G (say by an automatic break); when the contact is made the current flows through the coil and through BF, when it is broken the electromotive force of induction added to that of the battery enables'the current to pass through the cell and liberate the ions. At the make there is no such effect; there results therefore continued chemical decomposition.

Edlund investigated the integral electromotive forces of self-induction at the opening and closing of a circuit, and showed that they are equal. His experimental arrangement is very ingenious :—

G (fig. 51) is a differential galvanometer, A a coil whose self-induction is to be examined, C a wire wound in a zig-zag so as to have no self-induction. The battery E is connected at B and D with the circuit composed of G, A, and C, so that the currents in BcdCD and B5αAD pass round the coils of G in opposite directions. The resistance C is so arranged that there is no deflection of the needle in G. If now the current be stopped by breaking the circuit EB atK, the electromotive force dμe to the self-induction of A causes an extra current to flow round the circuit Aα5BcdCD, traversing the coils of G in the same direction. W e therefore get a deflection D l . In a similar manner if we make contact at K we get another deflection D,, due to the starting of the current in A. There is no difficulty in showing that, if E l , E 2 be the time integrals of the electromotive force in the two cases, then

⅛ Pi E ~D t ’

One of the difficulties encountered in such experiment« is the increase of the electromotive force of the battery E when it is left open for a time; this causes the extra current at make to be greater than that at break. Rijke, who made experiments similar to those of Edlund, avoids this difficulty by circuiting the battery, when BK is broken, The case of two circuits of invariable form and position is of great interest, from the light it throws on the action of the induction coil. We shall suppose that we have no soft iron core, and that the break in the primary is instantaneous. The latter condition is very far from being realized in practice, even with the best arrangements, so that our case is an ideal one.

Let i and j be the current strengths in primary and secondary, R and S the resistances, L, M, N the coefficients of induction, E the electromotive force in the primary. The equations are l ⅛ +m s+ k ∙-b < 3r '∙ m Λ + 1, ≡ + s ∙'" 0 ( 38 >∙

It is easy, in the first place, to show that the whole amounts of electricity which traverse the secondary at make and break of the primary are equal but of opposite signs. In fact, if we integrate (38) from the. instant before make to a time when the induction currents both in primary and secondary have subsided, we get >-m « where I denotes the steady current in the primary. Similarly integrating over the break, we get

∕⅛= + ⅛ 1 I = ME J S SR

In the second place, if we assume the break instantaneous, we can find the initial value of the direct current in S. Thus integrate (38) from the instant before break to the time τ after it, τ being infinitely short compared with the duration of the induction currents, then

- Μ14- N∕ o + ⅛J^jdt = 0.

o

Now the last term may be neglected, because τ is infinitely small and j is not infinite, hence we have, for the initial value of/, , - m I me urn Jo N “NR (40) ·

It is very easy now to determine the farther course of the current in S. The equation for j reduces to

⅛÷⅛∙-o i and we get, using (40),

>⅛^ i * <“>·

The direct induced current (current at break), therefore, starts in our ideal case with an intensity which is to the intensity of the steady current in the primary as the coefficient of mutual induction of the coils is to the coefficient of self-induction of the secondary, and then dies away in a continuous manner like any other current left to itself in a circuit of given resistance and self-induction.

Since we have already given enough of these calculations to serve as a specimen, we content ourselves with stating the result for the current at make. Owing to the self-induction of R, l fcc., the current in R rises continuously from zero to the value I; the induced current in S therefore begins also from zero, rises to a maximum, and then dies away. The mathematical exρressionfor it contains, as might be expected, two exponential terms.

It is instructive, in connection with what has already been said concerning the electrokinetic energy of two moving circuits, to examine what becomes of the energy in the case of two fixed circuits of invariable form.

Equations (37) and (38) may be used if, for generality, F be written instead of 0 in (38), so that there is electromotive force (say of constant batteries) in both circuits. Multiplying (37) by i and (38) by J, adding, and integrating from the time before E and F begin to act to a time τ when the currents have all become steady, we get

J(Er - RP)Λ + ∕cF∕ - S∕)Λ -1(Lt s + 2Mij+NJ s ) . (42).

0 0

In words, the excess of the chemical energy exhausted in the batteries over the amount of energy which appears as heat in the circuits is ∣(Lι 2 + 2My + N√ 2 ), which we denote by K. Similar remarks to those made at p. 7G apply here. K is the amount of electrokiuetic energy stored up in the medium surrounding the circuits during the time that E and F are raising the currents against self and mutual induction.

If we integrate similarly over the break of both currents, we find the defect of the chemical energy exhausted under the heat evolved in the circuit to be again K. Much of the energy thus discharged from the system at break usually appears in the spark.

Electrical Oscillations.—Helmholtz seems to have been the first to conceive that the discharge of a condenser might consist of a backward and forward motion of the electricity between the coatings, or of a series of electric currents alternately in opposite directions. Sir William Thomson took up the subject independently, and investigated mathematically the conditions of the phenomenon.

Let q be the charge of the condenser at time t, C its capacity, E the difference of potentials between the armatures, i the curreùt in the wire connecting the armatures, R its resistance, L the coefficient of self-induction. Then we have 5 -ce. .--f, and L⅞ + Rt = E,

. d 2 q R dq 1 λ z λx ,e ∙ ’ di + LΛ + LC ? = 0 (4S >·

The solution of this equation is j=e- ’ (Aβ^ + Be→ u ), . . . . (44), where nι=- n= ∕ — — — .

m 2L ’ √ 4L 2 LC

A and B are constants to be deteπnined by the conditions q = Q and ⅛ = 0 when t = 0.

Two distinct cases arise.

(1.) Let R be greater than~f~ J then the exponentials in (44) are real, the discharge is continuous, all in one direction, and involves no essentially new features.

(2.) Let R be less thaτ^∕; then the appropriate form of the solution is

7=*c~(A cos n< + B sin nZ), where m has the same meaning as before, but n stands now for /1 R 2 j^q - ^4L 2 ’ Η we determine A and B by the initial conditions, we get i = e - ∙'^eosnZ + ^ i sinwzjQ .... (45).

The current is given by

*~NΕC e ~"* sιn (46 )∙

It follows from these equations that, when R t ^ ιe charge of one armature of the condenser passes through a series of oscillations. The different maxima are

_ _ ∙>W Smτr 3..-7Γ

Q, -Qe^ » , Q< - >∣ , Qi^~, &c., occurring at times λ ιr 2ιr 3ιr .

θ . -, —, — , &C.

η η n When the charge has any of these maximum values, the current is zero. The current maxima form a similar descending geometric series, the times of occurrence being

θ θ + τr 0 + 2π 0 + 3iγ n ’ ~n~ ’ n , n ’ c ' , where 0 is the acute angle tan 1 —, m

The interval between any positive and the next negative maximum, whether of charge or current, is therefore .

We need not insist on the evident importance of this result. Thomson, in his original paper, points out the various applications of which it is capable. He predicts the phenomena afterwards observed by Feddersen; in fact, he suggests the use of Wheatstone’s mirror to detect it. Its bearing on the anomalous magnetization of steel needles by jar discharges, and on the anomalous evolution of gas by statical discharges, when electrodes of small surface are used (in Wollaston’s manner), are also dwelt upon.

Several physicists have taken up the experimental investigation of this matter. Feddersen’s experiments realize the case above discussed, if we abstract the disturbance owing to the air interval, of the effect of which it is not easy to give an accurate account. Feddersen’s results are in good general agreement with theory. He finds, for instance, that the critical resistance at which the discharge begins to assume the oscillatory character varies inversely as the square root of the capacity of the battery from which the discharge is taken. A good account of the researches of Paalzow, Bernstein, and Blaserna, and of the older researches of Helmholtz, remarkable for tl⅜e use of his pendulum interruptor, will be found in Wiedemann, §§ 801, etc. Schiller, in a very interesting paper, describes a variety of measurements of the period of oscillation, and the damping of the alternating currents in a secondary coil, when the current of the primary is broken. By means of the pendulum interruptor of Helmholtz (for description of which see his paper) the primary is broken, and at a measured interval thereafter the secondary circuit, which contains a condenser and a Thomson’s electrometer, is also broken. The deflection of the electrometer indicates the charge of the condenser at the instant when the secondary is broken. The interval between two null points separated by a whole number of oscillations can thus be found, and hence the time of oscillation of the coil calculated. The agreement of Schiller’s results with calculation is very remarkable, and must be regarded as a highly satisfactory proof of the validity of the theoretical principles involved.

Induction in Masses of Metal and Magnetism of Potation.—Hitherto we have dealt only with linear circuits; but it is obvious that currents will also be induced in a mass of metal present in a varying magnetic field. If the variation of the field be due to relative motion between the mass of metal and the system to which the field is due, the electromagnetic action of the induced currents will oppose the motion. Mauy instances might be given of this principle. If a magnet be suspended over a copper disc, or, better still, in a small cavity inside a mass of copper, its vibrations are opposed by a force due to the induced currents which for small motions varies as the angular velocity of the needle. Accordingly, it comes much sooner to rest than it would do if suspended in the air at a distance from conducting masses; it moves beside the copper as if it were immersed in a viscous fluid. Pliicker suspended a cube of copper between the poles of a powerful electromagnet, and set it spinning about a vertical axis; directly the magnet was excited it stopped dead. Foucault arranged a flat copper disc between the flat poles of an electromagnet placed at such a distance apart as just to admit it between them. The disc was set in rotation by means of a driving gear. So long as the magnet was not excited, the driver had comparatively little work to do; but as soon as the magnet was excited, the work required to keep up any considerable velocity greatlv increased. The additional work thus expended appears in the heat developed in the disc by the induced currents. Tyndall demonstrates this very neatly by causing a small cylindrical vessel of thin copper filled with fusible metal to rotate between the poles of an electromagnet, when enough heat is quickly developed to melt the metal.

On the other hand, when a mass of metal moves in the neighbourhood of a magnet, the electromagnetic action of the induced currents will cause the magnet to move, if it be free to do so. Thus, if we suspend a magnet with its axis horizontal over a disc which can be set in rotation about a vertical axis, owing to the electromagnetic action of the induced current, the needle will tend to rotate in the same direction as the disc. If the velocity be great enough, or the needle be rendered astatic, it may be carried round and round continuously. This action was discovered by Arago, and excited the attention of many philosophers, till it was at last explained by Faraday (see Historical Sketch). Many of the observations made by Faraday’s predecessors, and some made by himself, are at once seen to support the conclusion that the phenomenon is simply a case of Lenz’s law. Thus Snow Harris found that the deflecting couple on a suspended needle varied approximately as the velocity of the disc directly, and as the square of the distance of the needle from the disc inversely. It was also found that the action of the disc was directly proportional to the conductivity of the metal of which it was made, an exception occurring in the case of iron, whose action was disproportionately great. Cutting radial slits in the disc diminished the action very much.

Besides the component tangential to the disc, it is found that there is a repulsive normal action on the pole of the magnet, and also a radial action, which may be towards or from the centre of the disc, according as the pole is nearer or farther from the centre of the disc. These actions look at first sight somewhat more difficult to explain; but a little consideration will show that the laws of induction account for these also.

Let us first suppose the induced currents to appear and die away instantly after the small motion of the disc which produces them (we may suppose the motion of the disc to take place by an infinite number of small jumps). Thus the currents of induction are obviously symmetrical with respect to the diameter through the fool of the perpendicular from the magnetic pole on the disc, and we may roughly represent the elec tromagnetic action by a magnet placed perpendicular to the diameter at a certain distance from the centre of the disc, its south pole pointing in the direction of the disc’s motion if the inducing pole be a north pole. Let OX (fig. 52)be the direction of the diameter in the same vertical plane as the pole, NS the representative magnet, OY being the direction of motion. By our present supposition the inducing pole M lies in the plane of ZOX, in which case it is obvious that the resultant action reduces to a tangential component T parallel to OY.

But, owing to the inductive action on each other of the currents in the disc, the induced currents do not rise and fall instantaneously, but endure for a sensible time. We may roughly represent the effect of thia by supposing the representative magnet NS curried onwards a little with the disc, or, which amounts to the same thing, we may suppose the pole M to lag a little behind at M' (lying, say, on MM’ perpendicular to ZOY. ) The action of N will now preponderate, and the resultant force on M' will be in the direction M ’ F. This force, when resolved parallel to OY, OZ, OX, gives a tangential component as before, a repulsive normal component, and a radial component, which will be directed to or from the centre of the disc, according as the representative magnet lies farther from or nearer to the centre of the disc than the foot of the perpendicular from M'.

The original explanation of rotation magnetism (Faraday, Exp. Res., 81, &c.) should be consulted by the reader who wishes to pursue the subject. An account of the researches of Nobili, Matteucci, and others will be found in Wiedemann, Bd. ii. § 860, Ac. The mathematical theory has been treated by Jochmann, who neglected the inductive action of the currents on each other {Crellds Journ., 1864; Pogg. Ann., 1864; also Wiedemann, l.cj). A complete theory of the induction of currents in a plane conducting sheet has been arrived at by Maxwell by means of an extremely elegant application of the method of images {Proc. R.S., 1872; also Electricity and Magnetism, vol. ii. §§ 668, 669).

On the Origin of Electromotive Force.

It remains for us now to view the transformations of energy which take place in the voltaic circuit from the other side, and to inquire whence comes the energy that is evolved in so many different forms by the electric current. Two distinct questions are here involved. First—What form of energy is being absorbed, and at what part of the circuit does the absorption take place? Secondly—Where is the electromotive force which drives the current situated?

To the first of these questions experiment has given, on • the whole, a very satisfactory answer. The electric circuit is, indeed, one of the best instances of the great law of conservation, which states that the appearance of energy anywhere is always accompanied by the disappearance somewhere of energy to an equal amount. No general discussion of this first question is necessary; it will be sufficient to indicate the application of the general principle when we deal with particular instances.

Unfortunately the answers, both experimental and theoretical, that have been at different times given to our second question, are not so concordant as could be desired. The reader is, therefore, cautioned against accepting without due examination anything that may be here advanced.

Perhaps the most general principle concerning the origin of electromotive force recognized by physicists of the present day is the following:—

When two diferent substances are in contact, there exists in general an electromotive force at the surface of separation, tending to displace electricity across that surface.

This electromotive force is commonly called the “electromotive force of contact,” or simply the “contact force.” In the particular case of two conductors in contact, the effect of this force would simply be to maintain a certain difference of potential between them.

Although the earliest known case of electrification—viz. amber rubbed with woollen cloth—is an instance in point, and although many experiments on electrification by the friction of different substances were made, yet this principle was not recognized fully till the experiments of Galvani and Volta directed the attention of men of science to the matter.

Volta was the first to demonstrate clearly the existence of the contact force in the case of metals. A simplified form of his fundamental experiment is the following. The upper and lower plates of a condensing electroscope (see above, p. 34) are made of different metals, say copper and zinc. Let the upper plate be laid upon the lower, and the metallic contact ensured by connecting them for an instant by means of a wire. If the upper plate be now lifted vertically upwards, the gold leaves of the electroscope diverge, indicating that the zinc plate is now positively electrified to a considerable potential. This is explained as being due to the contact force at the junction of the copper wire with the zinc plate, by virtue of which the zinc is at a higher potential than the copper. Suppose the upper plate to be connected with the earth, then if E be the contact force, the potential of the zinc plate is E. Now E is very small, but as soon as the upper plate is raised the potential of the lower plate is increased in the same ratio as its capacity is diminished; hence the divergence of the leaves. Volta found that he could arrange the metals in series, thus— [table]

such that, when any metal is placed in contact with one below it in the series, it takes a higher potential; and he found that the electromotive force between any two metals in the series is the sum of the electromotive forces between every adjacent intervening pair. Thus, if Zn!Pb denote the electromotive force from lead to zinc, we get from the above table,

Zn∣Pb=5, Pb∣Sn = l , Zn∣Sn=ZnlPb +PbjSn=6 , PbiCu=Pb ’ 1 Sn + Sn[Fe + Fe^Cu=6, and so on.

It follows from Volta’s law that, if a number of metals be connected up in series, the difference of potentials between the extreme metals is independent of the intermediate metals, and, in particular, is zero if the extreme metals be the same. We cannot, therefore, have a resultant electromotive force in a closed circuit consisting of metals merely. This is entirely in accordance with experiment, provided the temperature be the same everywhere.

While one party of physicists neglected or attempted to explain away Volta’s contact force, another took up the investigation, and endeavoured to obtain precise measurements of it in different cases. Careful experiments of this kind were made by Kohlrausch and Gerland, by a method due to the former.

A condenser is used whose plates are made of the metals to be tested, say zinc and platinum. The plates are first placed parallel to each other at a very small distance apart, and touched simultaneously with a wire (say of platinum). A difference of potentials is thereby established, so that if the potential of the Pt be zero that of the Zn is Zn'Pt. (Here we neglect the contact force between air and zinc and between air and platinum. No experimental proof that we know of has been given in support of this, see below, p. 85). In consequence of this difference of potentials the Zn plate becomes positively charged. The wire is now removed, the plates of the condenser separated to a considerable distance, and the Zn plate connected with one electrode of a Dellmann’s electrometer, the other electrode of which is connected to earth. The reading is proportional to the potential difference ZniPt increased in the ratio in which the capacity of the Zn plate has been decreased by the separation. Hence, if A be the reading,

Zn∣Pt = λA (1).

The condenser plates are now brought into their former position, and connected through a Daniell’s cell, consisting of a strip of zinc immersed in a porous vessel filled with zinc sulphate, which is itself immersed in a vessel containing copper sulphate, into which dips a strip of copper. In the first instance, the copper strip is connected with the zinc plate, and the zinc strip with the copper plate of the condenser. The difference between the potentials of the condenser plates is easily found by an application of Volta’s law to be D + Zn∣Pt, where D denotes the difference between the potentials of the two pieces of copper forming the terminals of a Daniell’s cell; hence if B be the electrometer reading, after removing the Daniell and separating the plates as before, we have

D + Zn∣Pt = λB (2).

If we connect up the Daniell the opposite way with the condenser, then we get a reading C, such that

D-Zn∣Pt=∙λC (3).

From (2) and (3) we get

Z∙'∣Pt-∣⅛D (4∣, which gives the contact force Zn∣Pt in terms of the electromotive force of a Daniell. From (1), (2), (3) we get

B-C≈2A, an identical relation which the observations ought to satisfy, and which, therefore, affords the means of testing their accuracy.

In this way Kohlrausch found for Zu' l Cu the value ∙48D, or in other words, that the contact force from copper to zinc is about equal to half the electromotive force of a Daniell’s cell. As an instance of the general nature of the results, we give two series of numbers from the observations of Kohlrausch. The contact force is between zinc and the metal mentioned in first column in each case, and Zn∣Cu is taken = 100. [table]

In the second set of experiments the metals were carefully cleaned, whereas in the first set they may have been a little oxidized. This may very well account for the differences, for Kohl rausch found oxidized zinc strongly negative to freshly cleaned zinc. In fact, he found ZnlZnO = about ’ 4Zn∣Cu.

In order to test Volta’s law, a further series of observations was made, giving the contact force between iron and several metals. The following table gives the results observed directly and calculated from the table last given :— [table]

It will be seen that, with the exception of the values for Fe∣Cu, the agreement is very fair.

It is not necessary to give here the results of Gerland and Hankel. The latter made a great number of very careful experiments. He showed that the results depend on the nature of the surface of the bodies, being different when the surface is filed and when it is polished with rouge or other powder. His tables also show the gradual change effected in the contact force when the plates are exposed to the action of the air.

According to Volta, the contact forces between metals and liquids are either very small, and do not follow the same law as the contact forcée between metals, or else are absolutely non-existent. Subsequent observers, however, demonstrated the existence of contact forces in this case also, but showed that they do not obey Volta’s law like the contact forces in the case of metals.

Becquerel placed the fluid to be examined in a capsule of the metal, say copper. The capsule was placed on the upper plate of a condenser consisting of two copper plates in connection with a gold-leaf electroscope. The fluid and the lower plate of the condenser were touched each with a finger for a short time, and then the upper plate was removed. The divergence of the gold leaves was taken to indicate the contact force. In this way Becquerel found that ziιu, copper, and platinum were mostly negative to alkaline solutions; but the metals were in general positive to concentrated sulphuric acid. It is obvious, however, that the result of the experiment is complicated by the contact of the hand with the liquid and with the metal of the condenser.

Similar objections apply to the results of Pfaff and Peele t.

Buff made the lower plate of his condenser of the metal to be examined, of zinc for example; upon this was laid a thin glass plate on which was spread a thin layer of the liquid to be examined, or a piece of filter paper soaked with it. A zinc wire was used to bring the liquid and the lower plate of the condenser into communication; this wire was then removed and the glass plate with the liquid lifted. The divergence of the leaves was taken to indicate the contact force between zinc and the liquid. Although this method is an improvement on the methods of Becquerel and Peclet, it is still unsatisfactory, owing to the presence of the glass.

The most extensive and at the same time most careful experiments at present on record are those of Hankel.

The fluid (L) to be examined was placed in a widemouthed funnel. The condenser was formed by the surface of the liquid and a copper plate, which could be placed parallel to it at a very short distance apart, and raised as usual. The stem of the funnel was bent at a right angle twice, and ended in a wider portion, into which dipped a strip of the metal (M) to be examined. M was connected for a moment by a platinum wire with the copper plate and also with the earth. The wire was then removed, the plate was raised, and its potential determined by means of HankeΓs dry pile electroscope. The reading (A) is proportional to Cu∣Pt + Pt M + ML, or, by Volta’s law, to Cu∣M + M L. Hence

Cn∣M + M∣L-λA .

In the next place, the funnel is emptied and a plate of the metal M placed on its mouth. The copper plate is lowered so that it is at the same distance as before, contact established by means of the platinum wire, and so on. The reading being B, we have

Cu∣M = λB .

The plate of M is replaced by a plate of zinc, and the experiment repeated, and we have, C being r⅛e third read i∏g,

Cu∣Zn = λC .

From these three results we get M∣L i ~A q Cu∣Zn .

It is not necessary to quote HankeΓs results here. The reader may refer to Wiedemann’s Galvanismus, or tc HankeΓs paper.

Sir William Thomson has given a new proof of the existence of Volta’s contact force as follows. A ring is formed, one-half of which is copper the other half zinc. This ring is placed horizontally, and a needle made of thin sheet metal is so balanced as to form a radius of the ring. If when the needle is unelectrified it be adjusted so as to be over the junction of the two metals, then, when it is positively electrified, it will deviate towards the copper, and when negatively electrified, towards the zinc. Again, if a whole, instead of a half needle as above, be suspended over a disc made of alternate quadrants of zinc and copper, or, better still, inside a flat cylindrical box constructed in a similar way, so that when the needle is unelectrified its axis coincides with one of the diameters in which the disc is divided, then when the needle is positively electrified it will take up a position such that its axis bisects the copper quadrants; if it be negatively electrified, its axis will bisect the zinc quadrants.

Thomson has also given an elegant demonstration of the contact force between copper and zinc by means of an apparatus which is a modification of his water-dropping apparatus. A copper funnel is placed in a cylinder of zinc, and drops copper filings at a point near the centre of the cylinder. The filings are charged negatively by induction as they fall, owing to the excess of the potential of the zinc cylinder over that of the copper. If therefore the filings be caught in an insulated metal can, they will communicate to it a continually increasing negative charge, while the zinc cylinder and the copper funnel will become charged more and more positively.

Thomson finds, in agreement with Kohlrausch, that, when the copper and zinc are bright, the electromotive force of contact is about half that of a Daniell’s cell. When the copper is oxidized by heating in air, the contact force is equal to a Daniell or more. He has gone a step farther, and shown that when two bright pieces of copper and zinc are connected by a drop of distilled water, their potentials are as nearly as can be observed the same.

The subject of contact electricity has been taken up quite recently by Clifton. He experiments on the contact force between a metal and a liquid by a method which is a simplification o f Tlankel’s.

Two horizontal plates are used of the metal M; the liquid L is placed in a glass vessel on the lower plate and connected with the lower plate by a strip of the same metal which dips into it. The upper plate is lowered to a distance of 0 - l or 0 ’ 2 mm. from the surface of the liquid, which acts as the lower surface of the condenser, and the upper plate and lower plate are connected by a copper wire. The difference of potential between the two surfaces of the condenser is therefore M∣L. The copper wire is then removed, the upper plate raised, and its potential measured with a Thomson’s electrometer. In this way a contact force equal to the thousandth part of Zn∣Cu can be detected.

Clifton finds zinc and copper to be both positive to water to about the same degree, and both very slightly negative to dilute sulphuric acid. He concludes therefore that zinc and copper dipping in water will be at the same potential. This he verifies directly, finding that any difference of potential, if it exist, must be less than Ό0079 of the electromotive force of a Daniell. The result of Sir William Thomson is therefore confirmed.

There are many other points of interest in Clifton’s paper, but, as the results are given in most instances as preliminary, we need not discuss the matter farther.

Before leaving this subject, it may be well to notice that there is one point which is not touched by all these experiments, viz., the question whether there is or is not a contact force between metals or even liquids and air. It has not yet been shown that the results of the experiments which are supposed to demonstrate that Zn{Cu is about half the electromotive force of a Daniell could not be equally well explained by supposing the difference of potential to be Cu A + A Zn + CujZn, whence CffZn is very small compared with Cu'A and A∣Zn. This supposition would not invalidate Volta’s law; nor would it contradict Clifton’s results, for we have, in accordance with his experiments, on the new hypothesis,

AqlA + Cu∣Aq + AqJCu = Aq] A + Zn∣Aq + A Zn, therefore, transposing,

Zn∣Aq + Aq∣Cu + Cu',A + A Zn = 0 , which, according to the new hypothesis, means that copper and zinc immersed in water are at the same potential. In this view, the important part of the contact force usually observed between zinc and copper would be Cu∣A + A∣Zn, which must therefore, by Sir Win. Thomson’s result, be the same as Cu∣Aq + Aq'Zn.

It is not very easy to see how this point is to be settled by direct measurements of electromotive force. Supposing, however, that it were settled, and that the contact force between two given metals A and B, and between each of them and a given liquid L, were known, then the difference of potentials between the two metals when immersed in any liquid could be predicted in all cases, and also the initial electromotive force tending to send a current through a circuit made by connecting the metals together and dipping them into the liquid.

A number of cases of this kind have been investigated by Gerland; but satisfactory agreement between theory and observation has been attained in but a few cases. Researches of this kind are beset with a double array of almost insuperable difficulties.

The direction of the initial resultant electromotive force in any circuit of two metals and one fluid may be found by observing the first swing of a galvanometer through which the circuit is suddenly connected. Considerable precautions must be taken to obtain consistent results, and when all care has been taken, it is not found that the results of one observer always agree with those of another. This is scarcely to be wondered at, when we consider the difficulty of making sure that in two different experiments we are operating with absolutely the same metals and fluid in absolutely the same conditions as to surface.

When the current tends to pass from one metal to another through the liquid in which they are immersed, the former metal is said to be positive to the latter. If the whole electromotive force in the circuit be the sum of all the contact forces at the various junctions, then it follows easily that we ought to be able to arrange the metals in a series, such that any one in the series is positive to any following one and negative to any preceding when both are dipped in the same liquid. It does not follow that the order of the series is the same for different liquids; this would be so if the metallic contacts alone were operative.

Many electromotive series of this kind have been given by different experimenters; but they have lost much of their interest now that the theory of metallic contact, pure and simple, is given up. The following set is given by Faraday :— [table]

It will be seen that the order of the metals is not the same for different liquids.

Just as between different metals and between metals and liquids there is a contact force, so there is a contact force between different liquids. The direct observations of this contact force are very few and uncertain. One thing, however, is settled, viz., that the contact forces between liquids do not universally at least obey the law of Volta, or, at all events, do not form a consistent series with the metals; for a great variety of circuits of one metal and two solutions have been discovered in which the resultant initial electromotive force is not zero. Faraday has even found cases of this kind with one metal and two different strengths of the same solution.

The cell of Becquerel is a favourite illustration of such a circuit. It consists of a porous vessel filled with a solution of potash and immersed in a beaker containing nitric acid; two strips of platinum immersed in the potash and nitric acid respectively form the plates. The current goes in the cell from the potash to the nitric acid. The following additional examples are taken from Wiedemann.

The current flows from the metd through the liquids in the order named to the same metal again. For brevity, the metal is named only once. [table]

L stands for Zn, Cu, or Pt.

.M „ Cu, Fe, Pb, Sn, or Ag.

R ,, Ni, Bi, Pt, Hg, Pd, Sb, Fe, C, Ag, Zn, Cu, Cd, or Sn.

A great variety of active voltaic circuits have been formed with two liquids and two metals. The best known class of cases is that in which the metals are in contact, as in the two-fluid batteries of Daniell, Grove, and Bunsen. But Faraday gives a list of some thirty case3 in which the fluids and metals are placed alternately, so that there is no metallic contact. He marks the following combinations as powerful :— [table]

It must be carefully noticed that the galvanometer indi cation in the first instant only is to be considered. After the first rush of electricity the direction even of the current may alter. Above all, no conclusion concerning the value of the initial electromotive force is to be drawn from measurements of the subsequent current. Quantitative determinations of the electromotive force in many of the above cases have been made by various methods, of which an account will be found in Wiedemann’s Galvanismus, Bd. i. § 230. The most convenient plan is to use Thomson’s quadrant electrometer, Lippmann’s capillary electrometer, or some other instrument which allows us to measure the electromotive force while no current is passing through the cell. The galvanometer may also be used as in Latimer Clark’s modification of Poggendorff’s compensation method.

The apparatus may be arranged according to the scheme in fig. 53. ABC denotes part of the resistance in the circuit of the battery K; the circuits A/AeLB, AyFMC each contain a galvanometer, a cell, and a key. The cells E and F are so arranged as to tend to send currents in the same directions as K, but the resistances AB, AC are so adjusted that when the key L or the key M is depressed, no current is indicated by p or q. When this is so, we must obviously have E = Va - V b , F = V a - V c , <fcc., V a , V b , V c denoting the potentials at A, B, and C. Hence, if P, Q, R denote the resistances in AB, AC, and in the whole circuit of K, E -⅛K, F=«K.

If K were a constant battery, and its internal resistance were either known or else so small as to form only an inappreciable fraction of R, then each of the equations just written might be used singly, and we might operate with one cell and one galvanometer, comparing the electromotive force of the cell with K. In general, however, this will not be possible, and then we have, eliminating K and R,

E _ P

F - Q’ from which we get the ratio of E to F independent of the variation of K and R. We can by this method therefore find the ratio of the electromotive force of a given combination to that of a standard cell, when no current is passing through either. The process would be perfect in practice if a standard cell could be constructed whose absolute constancy could be relied on.

Contact force from polarization.—The flow of electricity through the cell is accompanied by a deposition of the products of chemical decomposition on the platesj which alters the surface contact forces. This constitutes the phenomenon of polarization, which we have already partially studied. It will be useful to consider a little more in detail some of the forms in which it is met with.

The products of electrolysis which accumulate at the electrodes may be simply held in solution or precipitated, or they may combine chemically with the solution; they may be deposited as a crust on the electrode, or they may enter into more or less intimate connection with it. Several of these different effects may occur together; but in almost all cases the result is the same, viz. a great weakening of the current after the first instant or so. This weakening of the current might be due either to a transition resistance caused by the alterations at the electrodes, or to an op·

posing electromotive force arising from the alteration of the contacts. The former was the explanation adopted in the earlier researches of Poggendorff and Fechner; but there can be no doubt about the existence of the latter effect, and Lenz showed that it was sufficient to account for the facts observed. It has been usual, therefore, to neglect the transition resistance altogether in the great majority of cases. It is certain, however, that it really exists in some instances. Consider the case of two electrolytic cells containing concentrated sulphuric acid, the electrodes being in the one copper plates, in the other platinum plates. Either of these cells inserted into a voltaic current will weaken the current, but for different reasons. In both cases hydrogen is freed at the negative electrode, and reduces sulphur from the strong acid, the effect of which is not great either way, for if the negative electrode be replaced by a fresh plate the weakening of the current remains. At the positive electrode oxygen is evolved, which oxidizes the copper in the one case and is deposited on the platinum in the other,—in both cases replacing the positive electrode by a fresh plate will cause momentary increase in the current; but the copper plate which served as positive electrode if tested against a fresh copper plate gives very little return current, whereas the positive platinum plate similarly tested gives a very powerful one. In the one cell the weakening of the current was due to the resistance of a crust of non-conducting copper oxide, in the other it was due to the contact force at the surface of the oxygenated platinum.

The polarization which arises from the deposition of gas on the electrodes is, if we except the case where peroxides are formed, by far the most powerful form, and has been much studied ever since voltaic batteries were used. Experiments like the one just quoted prove decisively that the electromotive force has its seat at the surface of the electrode itself, and is due to local alterations there. The certainty of this fact gives the study of the phenomenon great theoretical importance, since we may hope thereby to arrive at some idea of the nature of the contact force.

It is also certain that in most cases each electrode contributes separately to the whole electromotive force, for if we remove the polarized plates the adhering gas goes with them, and each plate is found to be electrically different when tested against a fresh or unpolarized plate. We may also take measures to remove the deposited gas by washing the plates with water, potash, or nitric acid, or by iguiting them; and it is found that the more energetic the chemical agency thus employed the more thoroughly is the polarization destroyed.

It seems clear, therefore, that it is the mere fact of the presence of the gas on the electrode in some form or other which causes the electrical difference. We may go still further and produce the phenomenon by depositing gas by means other than electrolytic. If a piece of platinum foil be immersed in hydrogen it absorbs the gas, as has been shown by Graham. A piece of foil thus treated is positive to a piece of freshly ignited foil when both are placed in dilute sulphuric acid. The same result is obtained by saturating the liquid in the neighbourhood of the foil with hydrogen.’ The activity thus conferred on the plate may be again destroyed by immersing it in chlorine or bromine, or even in oxygen, by igniting it, and so on. Similarly, if we dip a fresh piece of foil into chlorine or bromine, it will become negative to a fresh plate. The effect obtained by dipping the foil in oxygen in the ordinary state is very small, the oxygen deposited by electrolysis must therefore be in a different state to that condensed during mere im mersion in the gas. This is probably due to the fact that electrolytically generated oxygen contains ozone (see art. Electrolysis ); and in accordance with this we find that a platinum foil ozonized by being held in the electric brush proceeding from a charged conductor, or rubbed with phosphorus, is negative to a fresh plate in dilute sulphuric acid.

The gas battery of Grove is a remarkable instance of the electro- ’ motive properties of gas-coated metals. Two long glass tubes A j and B are arranged in the two necks of a Woulfe’s bottle. The upper ends of the tubes are closed, but pierced by two platinum wires, to which are fastened two long strips of platinum foil (which are sometimes platinized) reaching to very near the lower ends of the tubes. The bottle and part of the tubes are filled with some liquid, say dilute sulphuric acid, and hydrogen introduced into B and oxygen into A. This may be very conveniently done by sending an electric current from A to B and decomposing the dilute acid, but it may be done in other ways as well. This arrangement has an electromotive force comparable with that of a Daniell’s cell (see below, p. 88), and if the original volume of hydrogen in B be twice that of the oxygen in A it will continue to send the current through a closed circuit, the gas gradually disappearing in the tubes until none is left, when the current stops. These gas elements may be connected up in series, &c., and used like ordinary batteιy cells.

If the tube B be filled with ordinary hydrogen and A with liquid only, a current in the same direction as before is observed; but the liquid in A is decomposed and hydrogen evolved, which produces an opposing electromotive force and stops the current. If A contain oxygen and B fluid only, the current lasts for a very short time unless the oxygen contain ozone. This is in accordance with what we have already seen.

Grove has given an electromotive series for the different gases and metals as follows:— [table]

We have already remarked that the polarization in many cases comes on very rapidly. In cases where the electromotive force of the battery is not sufficient to produce a continuous evolution of gas, the current after the first rush dies away very rapidly. There are cases, however, in which a small current continues to flow for a very long time. Such currents are not accompanied by any visible evolution of gas, and it is clear that they could not be so accompanied, for, if the electromotive force of the battery be under a certain limit, the amount of chemical energy absorbed by the current per unit of time is less than the intrinsic energy of the ions liberated in a unit of time. It was originally supposed, therefore, that, besides this electrolytic conduction proper, fluids conducted to a slight extent like metals. But Helmholtz has shown that no such assumption is necessary, and has pointed out that when the fluid holds gas in solution a sort of electrolytic conduction may take place which involves, it is true, liberation of the ions, or at least of an ion, but so that the final result does not imply absorption of more energy than the battery can furnish per unit of time in accordance with Faraday’s law of electrolytic conduction.

Suppose, for instance, that the dilute sulphuric acid in an ordinary voltameter held II in solution. When the current passes, 0 appears at the anode and unites with the H in solution to form water; a corresponding quantity of H is thereby liberated at the cathode, which is either absorbed by the platinum electrode or diffuses towards the anode, to combine in its turn with the appearing oxygen. It is obvious that the liberation of the ion in such a case does not involve absorption of energy to the extent necessary when both H and 0 are disengaged from water. A current might therefore be kept up under such circumstances for a long time by an electromotive force under that of a cell of Daniell. Helmholtz has given the name of electrolytic convection to this species of electrolytic conduction. He has shown that the phenomenon comes to an end when the liquid is perfectly freed from gas. The absorption of the gases by the electrodes plays a great part here, and gives rise in gas-free liquids to a phenomenon analogous to the residual discharge. When the battery is first turned on, a rush of electricity takes place, then there is a small current which gradually dies away. The first rush is like the instantaneous charge of a condenser; the small current which arises from the slow penetration of the ions into the platinum corresponds to the formation of latent charge. When the voltameter is disconnected from the battery and discharged through a galvanometer, we have a first rush due to the part of the ions accumulated on the surface of the platinum, and then a gradually decreasing current due to the emergence of the gas which had penetrated into the metal. When the electromotive force which presses the gas into the electrode is removed, the absorbed gas will move very nearly in accordance with the ordinary law of diffusion, and the rate of its reappearance will depend very little on the electromotive force at the surface of the electrode. Consequently the residual current furnished by such an apparatus will not depend on its external resistance. Λ sudden increase of the external resistance will simply cause the current to diminish until sufficient surface density has been attained to raise the electromotive force to the value required to keep up the same current as before through the increased resistance; and the converse will happen if tlιe external resistance be decreased.

This passage of the gas into the substance of the electrode has, at the instance of Helmholtz, lately been investigated by Root. He finds that in certain cases, when only one side of a platinum foil is exposed to electrolysis, the deposited gas, whether H or 0, actually passes through and produces the corresponding polarization on the other side of the foil.

It might at first sight be expected that in all cases where the electromotive force in the circuit is sufficient to pro duce continual evolution of the ions the polarization would be the same. This is not by any means the case, however, owing to the fact that the final state of the liberated ions varies with the strength of the current, or, more correctly speaking, with the current density, i.e., the amount of electricity which crosses unit section of the electrode in unit time. When H 2 and 0 are being liberated from dilute H 2 SO 4 , this depende mainly on the formation of variable quantities of ozone and H 2 O 2 . This variation of the physical state and intrinsic energy of the liberated ions, is a fact of the greatest importance in the art of electrometallurgy. A better instance could not be given than Gore’s electrolytic modification of antimony, whose intrinsic energy is strikingly manifested by its explosive properties.

The effect of enlarging the surface of the electrode in diminishing the polarization in the case where the maximum polarization is not reached was noticed above (p. 48). It has also the effect of diminishing the maximum of polarization in the case where the ions are completely set free. Platinizing has the same effect. Thus Poggendorff found for the maximum polarization 2T2 to 2’32 for bright platinum plates, while it was only 1’83 to 1 , 85 for platinized plates. The effect of platinizing on the hydrogen and oxygen polarization was about equal for strong currents, but greater on the hydrogen polarization when the current was weaker. On the other hand, by using small platinum points to decompose water in Wollaston’s manner, Buff found the polarization as high as 3 ’ 31.

Poggendorff and Crova have investigated the dependence of the maximum of polarization on the current density. It follows from their researches that it can be represented by A - B~ βl , I being the current density.

It would appear that the maximum of polarization is decreased by increasing the temperature of the cell. The effect, however, in some cases at least which have been adduced to prove this, might be explained by the decrease of the internal resistance of the cell.

Agitating the electrodes, stirring the liquid in their neighbourhood, or any other process which tends to dissipate the deposit ou the electrode, leads, as might be expected, to a diminution of the polarization. The effect of increased pressure in retarding or helping electrolysis might be appreciable in certain cases. Suppose that an electrochemical equivalent of the ions, during liberation under a pressure p, increases in volume by v, then during the passage of a unit of electricity work to the extent pv is done; the electromotive force required to free the ions must therefore have a part equal to pv which may increase or decrease as the process goes on. If the ions be gases which obey Boyle’s law very nearly, then pv is constant, so long as the temperature remains the same; so that we cannot expect, within reasonable limits, to check the electrolysis of dilute sulphuric acid by conducting it in a closed vessel.

We have repeatedly drawn attention to the rapidity with which the polarization decays in the first few instants after the plates are connected through a circuit of moderate resistance. Direct proofs of this have been given by Beetz and Edlund. The former shows that the oxygen polarization decays much more rapidly than the hydrogen polarization, which is not to be wondered at, considering the greater readiness of platinum to absorb hydrogen; with palladium electrodes the difference would doubtless be still more marked. The reader may also consult an interesting paper on this subject by Bernstein who concludes that in a certain case the polarization diminished by ⅜ to τ ⅛ of its value in about i ⅜ τy of a second.

There seems to be little reason to doubt the substantial accuracy of the facts just mentioned; and the reader will not fail to see the application to the theory and practice of the measurement of the electromotive forces of inconstant electromotors, a category under which, unfortunately for the electrician, all known voltaic batteries must be classed. The remark applies with double force to the measurement of the electromotive force of polarization. Many measurements of the latter have been made. We quote a few, to give the reader a general idea of the magnitudes involved; into a discussion of the methods we cannot enter here. [table] [table]

[table]

Although the polarization by gas deposits has absorbed so much of the attention of physicists, it is by no means a solitary instance. The phenomenon is universal. It appears even with zinc plates in zinc sulphate, and copper plates in copper sulphate. The nearest approach to unpo-larizable electrodes is the case of amalgamated zinc plates in zinc sulphate, originally discovered by Du Bois Reymond. When the sulphate solution is neutral, the polarization, as may be shown by immersing a large number of plates in series in the solution, is extremely small.

For an account of polarization at the surface of two liquids observed by Du Bois Reymond, and other kindred matters, and for many other facts which we have passed over in silence, the reader may consult Wiedemann’s Galvanismus. Some account will be found in the article Electrolysis of the remarkable phenomenon of the “passivity of iron, and of the powerful polarization arising from the formation of superoxides, on which depends the action of the secondary pile of Planté.”

Application of the Laws of Energy to the Voltaic Circuit. —In the classical series of researches by which Joule laid the foundations of the laws of energy, a considerable share of attention is devoted to the energetics of the electric current. Guided by the great idea which he was gradually developing, Joule made experimental determinations of the amount of energy of various kinds evolved in the electric circuit. We have already seen how he measured the quantity of heat developed in a metallic conductor, and in an electrolyte. This quantity was found to vary as the product of the resistance of the conductor into the square of the current strength, account being taken of disturbances at the electrodes in the case of electrolytes.

These disturbances were considered in the first memoir and allowed for. The accuracy of the view taken of them, to which Joule was led by the opinion of Faraday, that the solution of the oxide in the voltaic cell had no active share in producing the electric current, was justly questioned, implicitly by Sir Win. Thomson in 1851, and explicitly by Bosscha in 1859.

In a later memoir, however, Joule made a direct experimental investigation of these secondary effects, and shows how they can be accounted for. His results have not been shaken by subsequent investigators; and the general conclusions to be drawn from them are not in the least affected by the theory of secondary action, which is suggested in the paper. These, so far as we are now concerned with them, are as follows :—

“1st. In an electrolytic cell there are three distinct obstacles to the voltaic current. The first is resistance to conduction; the second is resistance to electrolysis without chemical change [arising simply from the presence of chemical repulsion]; and the third is resistance to electrolysis, accompanied by chemical changes.

“2d. By the first of these (the resistance to conduction) heat is evolved exactly as it is by a wire, according to the resistance and the square of the current; and it is thus that a part of the heat belonging to the chemical actions of the battery is evolved. By the second a reaction on the intensity of the battery occurs, and wherever it exists heat is evolved exactly equivalent to the loss of heating power in the battery arising from its diminished intensity. But the third resistance differs from the second, inasmuch as the heat due to its reaction is rendered latent, and thus lost to the circuit.

“3d. Hence it is that, however we arrange the voltaic apparatus, and whatever cells of electrolysis we include in the circuit, the whole caloric of the circuit is exactly accounted for by the whole of the chemical changes.

“4th. As was discovered by Faraday, the quantity of current electricity depends upon the number of atoms which suffer electrolysis in each cell; and the intensity depends on the sum of chemical affinities. Now both the mechanical and heating powers of a current are (per equivalent of electrolysis in any one of the battery cells) proportional to its intensity. Therefore the mechanical and heating powers of the current are proportional to each other.

“5th. The magnetic electrical machine enables us to convert mechanical power into heat by means of the electric currents which are induced by it; and I have little doubt that, by interposing an electromagnetic engine in the circuit of a battery, a diminution of the heat evolved per equivalent of chemical change would be the consequence, and this in proportion to the mechanical powers obtained.”

The above statement of Joule’s contains, in a form which seems to us neither ambiguous nor obscure, an exposition of the leading experimental principles of the energetics of the electric circuit. Besides the paj>ers of Joule we have mentioned, two others on the electrical origin of the heat of chemical combination ought to be read in connection with this subject. The now famous tract of Helmholtz, “Ueber die Erhaltung der Kraft,” which appeared in 1847, shortly after these papers of Joule, did much, by its able statement of the issues, to advance this branch of electrical science, and should be consulted by every thorough student.

An extremely important contribution to the experimental evidence for the law of energy in the case of electric currents was furnished by the researches of Favre. He uses a calorimeter, which is virtually a mercury thermometer with an enormous bulb, into which are inserted a number of test-tube shaped vessels all opening outwards. When a heated body is placed in one of these vessels its heat is quickly communicated to the mercury in the calorimeter, and the amount of heat thus communicated is measured by the expansion of the mercury, which is measured as usual by noting the displacement along a capillary tube. Into one of the recesses of the bulb of this calorimeter containing a quantity of dilute sulphuric acid was introduced 33 grm. of granulated zinc. The heat evolved during its dissolution was 18682 units (gramme-degrees C.). Five of the recesses were then furnished with dilute sulphuric acid of the same strength as before, and into them were put five elements of Smee (amalgamated zinc and platinized copper). These were joined up in circuit by means of very thick copper wire, and the heat developed during the solution of 33 grm. of zinc observed as before. The result was 18674 units, t.e., almost exactly the same as before. A small electromagnetic engine was next introduced into the circuit, and the heat observed, first, when it was at rest; secondly, when it was in motion, but consuming all its energy in heat owing to friction, &c.; and thirdly, when it was doing work in raising a weight. The quantities of heat in the three cases were 18667, 18657, and 18374 units respectively. In the first four experiments, therefore, the heat developed in the circuit is sensibly the same, the mean being 18670; the heat developed in the last case is less than this by 296, which is the equivalent of the potential energy conferred on the raised weight. From this result the value of the mechanical equivalent of heat ought to be 443. This differs considerably from the best value (423 to 425), but not more so than might be expected from experimental errors.

Dynamical Theory of the Electromotive Force of the Battery.—In two very important papers published in the Philosophical Magazine for 1851, Sir William Thomson laid the foundations of the Dynamical Theory of Electrolysis, one of the objects of which, to use very nearly his own words, is to investigate, for any circuit, the relation between the electromotive force, the electrochemical equivalents of the substances operated on, and the dynamical equivalent of the chemical effect produced in the consumption of a given amount of the materials, and by means of this relation to determine in absolute measure from experimental data the electromotive force of a single cell of Daniell’s battery, and the electromotive force required for the electrolysis of water.

The relation sought for is found as follows. Let E be the electromotive force of the battery. Then, by the definition of electromotive force, E is the whole work done in the circuit by a unit current during a unit of time. This work may appear as heat developed in the conductors or at the junctions according to the laws of Joule and Peltier, as the intrinsic energy of liberated or deposited ions, as work done by electromagnetic forces, as “local heat” in the battery (see below, p. 91), or otherwise. Let e be the electrochemical equivalent of any one of the elements which take part in the chemical action from which the energy of the current is derived, i.e., the number of grammes of that element which enter into the chemical action during the passage of unit current for a second; and let θ be the thermal equivalent of that amount of chemical action in the battery into which exactly a gramme of the element in question would enter,—in other words, the amount of heat that would be developed were the whole energy of the amount of chemical action just indicated spent in heat. Then it is obvious that the energy of the chemical action that takes place in the battery during the passage of unit current for a unit of time is JeØ, where J is Joule’s equivalent. Hence, by the principle of conservation, we must have

E = Jeθ; or, in words, the electromotive force of any electrochemical apparatus is, in absolute measure, equal to the dynamical equivalent of the chemical action that takes place during the passage of unit current for a unit of time.

The value of J is known, being 4156 × 10 4 in absolute units. The value of e has been found by various experimenters (see below, p. 104), the most accurate result being probably that deduced from the experiments of Kohlrausch, viz. <? = Ό03411 for zinc.

We may find θ by direct calorimetric experiments on the heat developed in the circuit. In this way Joule found for the thermal equivalent of the chemical action of a Daniell’s cell during the solution of 65 grammes of zinc 47670 (grm. deg. C.), and Raoult, by a somewhat similar process, obtained the number 47800. These give for the heat equivalent of the chemical action during the solution of 1 grm. zinc 733 and 735 respectively. Substituting these values in our formula, we get for the electromotive force of Daniell’s cell in absolute C. G. S. units 1039 × 10 8 or D042× 10 β .

But we may proceed in a totally different way to find the value of θ. Direct observations have been made on the heat evolved iu a great variety of chemical actions, and from these experiments we can calculate, with a considerable degree of accuracy, the value of θ, and thus deduce the electromotive force of a battery from purely chemical data. This method of procedure must of course be adopted if we wish to test the dynamical theory. Now, neglecting refinements concerning the state in which the copper is deposited, the state of concentration of the solution, &c., the chemical action in a Daniell’s cell may be stated to be the removal of the Cu from CuSO 4 in solution, and the substitution of Zn in its place. Now, Favre and Silberman have found for the heat developed in this chemical action 714 (grm. deg. C.) per grm. of zine. This will give, by the above formula, for the electromotive force of Daniell’s element 1O12×10 κ . The chemical action might also be stated as the combination of zinc with oxygen, and the solution of the zinc oxide thus formed in sulphuric acid, the separation of copper oxide from sulphuric acid, and of the copper from the oxygen. The quantity of heat evolved in the first two actions per grm. of zinc is, according to Andrews, 1301 +369 = 1670 (grm. deg. C.), and that absorbed in the last two actions 588 + 293 = 881. Hence 0 = 789; this gives DI 18 × 10 8 . Professor G. C. Foster has calcιdated from Julius Thomsen’s experiments values 805, 1387, and 617 of θ for the cells of Daniell, Grove, and Smee respectively; the values of the electromotive forces corresponding to these are U141×10 8 , 1 , 966 ×10 8 , and - 875 × 10 8 . These results are in fair agreement with the different values of the electromotive force obtained from direct experiment.

It follows from Thomson’s theory that a certain mini mum electromotive force is necessary to decompose water; and he calculated from the data of Joule that this minimum was U318 times the electromotive force of a Daniell’s cell. Favre and Silberman found for the heat developed in the formation of H 2 O 68920, from which we conclude that the minimum electromotive force required to electrolyse water is 68920 ÷ 47800, i.e., L44 times that of a Daniell’s cell.

Development of Heat at the Electrodes.—In a remarkable paper, which we have already quoted, Joule investigated directly the disturbing effect of the electrodes on the heat developed in an electrolyte. His method was as follows. A coil of wire whose resistance was known in terms of a certain standard was inserted in the circuit of six Daniell’s elements, and the heat evolved in it carefully measured by immersing it in a calorimeter. The resistance of the rest of the circuit, including that of the battery, was found by interpolating a known resistance in the circuit and observing, by means of a tangent galvanometer, the ratio in which the current was reduced. (The assumption is here made that the electromotive force of a Daniell’s cell is constant for different currents.) Knowing the heat evolved in a part of the circuit of known resistance, and knowing the resistance of the whole circuit, the heat evolved according to Joule’s law in the whole circuit during the oxidation of 65 grammes of zinc can be calculated from the indications of the tangent galvanometer previously compared with a voltameter. Hence the thermal equivalent Θ of the work done by the electromotive force of a Daniell’s cell during the solution of 65 grm. zinc can be found. The value of Θ arrived at by Joule in this way is 47670 (grm. deg. C.).

Electrolytic cells of various construction were then inserted into the circuit. The electromotive force resisting the passage of the current through the cells was found by taking the number of battery cells just sufficient to produce electrolysis, observing the current, then increasing the number of cells by one and observing the current again. If i be the current in the first case, corrected to bring it to the value it would have had if the resistance of the whole circuit had been the same as in the second case, and j the current in the second case, then, E being the number of battery cells used in the first case, the electromotive force Z opposing the current is given by

Z = E-U., I-J

the unit being the electromotive force of a Daniell’s cell. Z being known aud assumed constant for different currents within certain limits, the resistance of the whole circuit, electrolyte included, can be found by Ohm’s method as above. The amount of heat H which ought to be developed in the electrolyte by Joule’s law can then be calculated. The amount of heat II' actually developed was observed. It was found that II' is in general greater thau H, the difference per electrolysis of 65 grm. zinc with various electrodes is shown in the following table:— [table]

The electrolyte in all these cases was dil. H.,SO 4 , excepting the last case, where it was a solution of potash of sp. g. 1Ό63. In all the cases the chemical process is finally the same or very nearly so, viz., the freeing of the elements of water, hydrogen and oxygen, in the ordinary gaseous state, and the transference of a certain quantity of H.,SO 4 from the negative to the positive electrode, or of alkali in the opposite direction. Yet the values of H' - H (which we may call the local heat) are very different. It will be seen, however, that the values of H' - H and Z rise and fall together; and, if we calculate the electromotive forces (L) corresponding to thevalues of H' - H, by dividing by 47670, which was found for the thermal equivalent of the electro motive force of a Daniell’s cell, and subtract the values of L thus found from Z, we get results which are not far from constant. The mean of the values of Z - L is 1 ’ 36, the thermal equivalent of which is 64800, which is not very different from 68900, the heat evolved in the combination of 2 grm. of H with 16 grm. of O to form water. It appears, therefore, that the local heat corresponds to the excess of the electromotive force of polarization over the electromotive force requisite to separate water into its component gases. In other words, the work done by the current against this residual electromotive force is accounted for by the local heat H' - H developed in the cell (see Joule’s statement above, p. 89). A glance at the column L in the above table shows that this residual electromotive force depends greatly on the nature of the electrodes. Thus when the positive and negative electrodes are plates of platinum and zinc respectively the residual electromotive force is D39, whereas with platinized silver plates it is only , 34. Local heat and the corresponding electromotive force play a very important part in the working of batteries. Owing to this cause there is an evolution of heat in the cell itself which varies with tire strength of the current, and uses up a certain definite fraction of the energy furnished by the solution of the zinc. By sufficiently increasing the external resistance, the amount of heat developed in the cell according to the law JH = RI can be made as small a fraction as we please of the whole heat thus developed; but the amount of local heat generated in the cell during the solution of 65 grm. zinc is not greatly altered in this way, at least not in a cell of Daniell, or in any other of the so-called constant batteries. Did our space permit we might quote a great variety of experimental results to illustrate the principles we have been discussing. Most of these investigations are due to the French physicists Favre and Silbermann, whose researches have greatly advanced this department of the science of energy.

Very copious extracts from the memoirs of these and other physicists who have worked in this department will be found in Wiedemann, Bd. ii. 2, §§ 1121 sqq. The reader who desires to follow this interesting subject to the sources will find his labour much lightened by referring to M. J. Violle’s excellent bibliography of the mechanical theory of heat, appended to the second volume of Verdet’s Theorie Mécanique de la Chaleur. Much has been done for the theory of the subject by a series of papers on the mechanical theory of electrolysis by Bosscha, 2 in which the somewhat complicated phenomena involved are analysed in a remarkably clear and able way. Any reader who desires to know what has been done in this department will do well to consult these papers. We quote the following as an example of Favre and Silbermann’s results and of the calculations of Bosscha.

The heat evolved in a cell of Smee and in platinum wires of different lengths through which it was circuited was measured with the following result :— [table]

The heat in each case is that evolved during the liberation of 1 grm. of hydrogen in the cell. If we assume that the whole heat in the cell and in the wire is generated according to Joule’s law, and calculate on this hypothesis the resistance of the cell in mm. of the wire, we should get values varying from 66 to 200 mm. If, however, we assume, in accordance with the principles explained above, that a constant fraction of the whole energy per grin, of liberated hydrogen appears as local heat in the cell, then, Q denoting the whole heat which appears in the cell, L the local heat, H the heat in the wire, R the resistance of the cell, S that of the wire, we have

Q-L R H S ; audit is found that on making R = 32 ’ 3 and L = 7589, this formula will represent the results of experiment very fairly. The last column in the above table gives the value of Q thus calculated. In general so good an agreement is not to be expected, because L may and does vary with the strength of the current.

i Thus far we have been dealing with the direct results of experiment, but when we inquire into the reason for the existence of this residual electromotive force and of the local development of heat corresponding to it, and, in particular, when we ask why the effect is so much greater with some metals than with others, the answers become less satisfactory. We meet, in fact, with considerable divergence of opinion.

Joule’s view was that the effect is due to the affinity of the metal of the electrode for oxygen. This is endorsed to a certain extent by Sir William Thomson, who puts the matter thus: —“In a pair consisting of zinc and tin the electromotive force has been found by Poggendorff to be only about half that of a pair consisting of zinc and copper, and consequently less than half that of a single cell of Smee’s. There is therefore an immense loss of mechanical effect in the external working of a battery composed of such elements, which must be compensated by heat produced within the cells. I believe, with Joule, that this compensating heat is produced at the surface of the tin in consequence of hydrogen being forced to bubble up from it, instead of the metal itself being allowed to combine with the oxygen of the water in contact with it. A most curious result of the theory of chemical resistance is that, in experiments (such as those of Faraday, Exp. Res., 1027, 1028) in which an electric current passing through a trough containing sulphuric acid is made to traverse a diaphragm of an oxidizable metal (zinc or tin) dissolving it on one side and evolving bubbles of hydrogen on the other, part (if not all) of the heat of combination will be evolved, not on the side on which the metal is being eaten away, but on the side at which the bubbles of hydrogen appear. It will be interesting to verify this conclusion by comparing the quantities of heat evolved in two equal and similar electrolytic cells in the same circuit, each with zinc for negative electrode, and one with zinc the other with platinum or platinized silver for the positive electrode.”

Bosscha diesents from the theory of “chemical resistance” thus expounded, and advances a different explanation. According to him, the local heat arises from the energy lost by the liberated ions in passing from the active to the ordinary state. We know that the hydrogen which is being liberated at the surface of an electrode can effect reductions which hydrogen in the ordinary state cannot accomplish; it is liberated in fact in a state of greater intrinsic energy than usual. It is this excess of intrinsic energy which appears as local heat, and gives rise to the residual electromotive force in electrolysis. Different metals possess in very different degrees the power of reducing active hydrogen to the ordinarv state: and therefore the proportion of hydrogen which gets away from the electrode in the active state differs according to circumstances. Bosscha’s theory is that it is the intrinsic energy thus carmed away from the electrode that appears as local heat. Similar remarks apply of course to oxygen, the active form of the gas being probably ozone. As a verification of the theory, the fact is cited that at the surface of a plate of carbon, which possesses in an eminent degree the power of reducing ozone to the ordinary state, the residual electromotive force and local heat are very small. At all events the theory of “chemical resistance” is held to be inadequate to explain the facts; for calculating from some results of his own, combined with those of Lenz and Saweljew, he finds for the residual electromotive force at electrodes of

Pt Fβ Cu Sn Hg Zn

•45 ∙49 '64 ∙86 120 1*20; from which it appears that the order of magnitude of the electromotive forces is not that of the affinities of the metals for oxygen.

Electrical Measure of Chemical Affinity.—In a paper sent to the French Academy to compete for a prize offered for the best essay on the heat of chemical combination, Joule elaborates an ingenious method for measuring chemical affinity. By direct observation it is ascertained how much heat is developed in a given time in a certain standard coil of wire, when it is traversed by a current whose strength is measured by means of a tangent galvanometer. Then three readings of the tangent galvanometer are taken—first, when the galvanometer alone is in circuit with the battery, secondly, when the standard coil is also inserted, thirdly, when an electrolytic cell is inserted instead of the coil. The amount of the ions liberated and the heat evolved in the cell during a given time is also observed in the last case. If A, B, C be the readings of the galvanometer in the three cases, and if x be the resistance of a metallic wire capable of retarding the current equally with the electrolytic eell, then we easily get, taking the resistance of the standard coil as unity,

(A-C)B X (A≡^B)C'

Now if the resistance x were put in the place of the electrolytic cell, the current would be the same; hence by Faraday’s law the amount of chemical energy absorbed in the battery would be the same. Also the heat evolved in the rest of the circuit, excluding x, would be the same. It follows, therefore, that the heat H which would be evolved in x is the equivalent of the whole energy given out in the electrolytic cell. If therefore we subtract from II the heat K which actually appears in the cell, the remainder ∏ - K is the heat equivalent of the intrinsic energy of the liberated ions; and, provided they appear finally in the “ordinary” condition, II - K is the heat which would be developed when they are allowed to combine.

In this way Joule found for the heat evolved in the combustion of 1 grm. of copper, zinc, and hydrogen respectively 594, 1185, 33553.

Galvanic Batteries.—It would be inconsistent with our general plan to attempt an exhaustive discussion of all the different electromotors which depend for their energy on chemical action. Wiedemann’s Galvanismus, or books on telegraphy and other arts in which electricity is applied to technical purposes, may be consulted by the reader who wishes for fuller information. A brief discussion of some typical batteries will, however, be useful, were it only to illustrate the principles we have just been laying down.

All the earlier batteries were one-fluid batteries. The plates usually consisted of ziuc and copper, and the • exciting fluid was in general sulphuric acid. Various improvements were made by Cruickshank, Wollaston, Hare, and others, in the way of rendering the battery more compact, and of reducing its internal resistance by enlarging the plates. Hare carried the last-mentioned improvement to great lengths; by winding up together in a spiral form sheets of copper and zinc, insulated from each other by pieces of wood, plates of 40 or more square feet surface were obtained. In this way the internal resistance was very much reduced, and powerful beating effects could be obtained. When small internal resistance is no object, the cells of the battery may be filled with sand or sawdust, moistened with the dilute acid. In this form the battery is more portable.

There are two capital defects to which all one-fluid batteries are more or less subject. In the first place, whether there is or is not external metallic connection between the plates, a certain amount of chemical action goes on at the surface of the zinc, which consumes the metal without aiding in the production of the current. To this is given the name of local action. It is supposed to arise from inequalities in the zinc, on account of which one portion of the metal is electropositive to a neighbouring portion; hence local currents arise causing an evolution of hydrogen at some places and solution of the zinc at others. In the second place, when the battery is in action, there is always an evolution of hydrogen at the copper or electronegative plate of the cell, a certain amount of which adheres to the plate and causes a strong electromotive force of polarization. The first of these evils is remedied to a great extent by amalgamating the zinc plate. We thus reduce the surface metal to a fluid condition everywhere, and thereby eliminate differences of hardness and softness, crystalline structure, and so on. The oldest method was to use zinc amalgam for the electronegative metal; but it is now universally the custom to amalgamate the surface of the zinc plates simply, which may be done by rubbing them with mercury under dilute sulphuric acid. No such effective cure has been found for the hydrogen polarization. Smee introduced the plan of using instead of the copper plates thin leaves of platinum or silver foil, which are platinized by the process already described (p. 87). This, in accordance with what we have already seen, diminishes the polarization. A similar result is obtained by using for the electronegative plate cast iron or graphite; the action of the latter is much improved by steeping it in nitric acid.

This last fact introduces us to a new principle for improving the action of batteries, viz., the use of an oxidizing agent to get rid of the hydrogen polarization. When the plates of a Smee’s battery have been exposed to the air for some time, it is always found that the current is much more energetic than usual just after the first immersion. The improvement is of course only temporary, for the stock of oxygen is soon exhausted, and on raising the plates and dipping them again immediately, the phenomenon does not appear. Davy found that dilute nitric acid acted better than dilute sulphuric acid as au exciting fluid, and the cause of this is the action of the HN0 3 on the hydrogen evolved at the copper plate. Good instances of this kind of action are furnished by the bichromate battery of Bunsen and the Léclanché cell, which occupy a sort of middle position between one and two fluid batteries.

The bichromate cell consists of an amalgamated zinc plate, usually suspended between two parallel carbon plates, so that it can be raised or depressed at pleasure. The bichromate solntion is made, according to Bunsen, by mixing 605 parts of water with 61 ’ 8 of potassium bichromate and 116 of sulphuric acid. The electromotive force of the bichromate cell is very great to start with (more than twice that of a Daniell’s cell), but it falls very quickly when the external resistance is small. The cell recovers pretty quickly however, and may be used with advantage where powerful currents of short duration are often wanted. In the cell of Léclanché the electronegative metal is replaced by a porous vessel filled with carbon and pounded peroxide of manganese. The exciting liquid used is chloride of ammonium. Chloride of zinc is formed at the zinc plate, and ammonia and hydrogen appear at the negative plate; the latter reduces the MnO j , so that H,0 and Mn s O s are formed, while the ammonia is partly dissolved and partly escapes. Thiβ element is tolerably constant if it be not used to produce very strong currents, but its great merit consists in being very permanent; for it will keep in condition for months with very little attention, furnishing a current now and then when wanted; hence its extensive use in working electric bells, railway signals, and so on.

It cannot be said that any of the batteries we have described, or in fact any battery on the one-fluid system, satisfies to any great extent the requirements of a constant electromotor, which are to give the same electromotive force whatever the external resistance, and to preserve that electromotive force unaltered for a considerable time. The best solution of the problem to construct a constant battery is the two-fluid principle invented by Daniell; and on the whole, the best application of that principle is the cell originally given by him. This consists essentially of a plate of copper immersed in a saturated solution of copper sulphate, and a plate of zinc immersed in dilute sulphuric acid or zinc sulphate, the copper solution being separated from the other by some kind of diaphragm, usually a porous vessel of unglazed earthenware. The chemical action consists of the solution of the zinc plate to form zinc sulphate, the formation of zinc sulphate at the diaphragm, and the deposition of copper at the copper plate; thus :—

Zn SO 4 Zn SCζZn ∣ SO∕Tι SO 4 Cu Cα

gives ZnSO 4 ZnSO 4 Zn ∣ SO 4 CuSO 4 CÜCÜ.

The evolution of hydrogen and the polarization arising therefrom are thus avoided.

A very common arrangement of this cell is a porous vessel, containing the copper plate and the sulphate of copper, with a small store of large crystals to keep the solution saturated. This vessel is dipped into another nearly filled with a semi-saturated solution of zinc sulphate, in which is placed the zinc plate. With a little care to keep the cell clean by occasionally removing some of the zinc solution and diluting to prevent incrustation, to feed the copper solution, so that it may not get weak and deposit hydrogen instead of copper on the copper plate, to keep down the level of the copper solution, which is constantly rising by osmose (see art. Electrolysis), and a few other obvious precautions, a battery of Daniell’s cells will furnish a very nearly constant current, and keep in order for a long time. It is necessary to keep the current going, otherwise the solutions diffuse through the porous vessel, the result of which is a deposit of copper on the zinc, and other mischiefs, which stop the action of the cell altogether.

Daniell’s element has been constructed in a great variety of forms, to suit various purposes. The sawdust Daniell, invented by Sir Wm. Thomson (1858), is very convenient when portability is desired. In this form the copper plate, soldered to a guttapercha covered wire, is placed at the bottom of a glass vessel and covered with crystals of copper sulphate; over these wet sawdust is sprinkled, and then more sawdust, moistened with a solution of sulphate of zinc, upon which is laid the zinc plate. The cell of Minotto is very similar to this.

In these batteries the sawdust takes the place of the porous diaphragm, and retards the interdiffusion of the fluids. In another class of batteries, of which the element of Meidinger may be taken as the type, the diaphragm is dispensed with altogether, and the action of gravity alone retards the diffusion. In Meidinger’s cell the zinc is formed into a ring, which fits the upper part of a glass beaker filled with zinc sulphate. At the bottom of this beaker is placed a smaller beaker, in which stands a ring of copper, with a properly insulated connecting wire. The mouth of the beaker is closed by a lid, with a hole in the centre, through which passes the long tapering neck of a glass balloon, which is filled with crystals of copper sulphate; the narrow end of this neck dips into the small beaker, the copper sulphate runs slowly out, and being speciiically heavier than the zinc sulphate, it collects at the bottom about the copper ring.

Yet another form of Daniell’s element is the tray cell of Sir William Thomson, which consists of a large wooden tray lined with lead, the bottom of which is covered with copper by electrotyping. The zinc is made like a grating, to allow the gas to escape, and is enveloped in a piece of parchment paper bent into a trayshape, the whole resting on little pieces of wood placed on the leaden bottom of the outer tray. Sulphate of copper is fed in at the edge of the tray, and sulphate of zinc is poured into the parchment. The zincs in these elements are some 40 centimetres square, so that the internal resistance is as low as 0’2 ohm.

One of the best known in this country, and perhaps the most . used of all the two-fluid cells, is the element of Grove. This differs from Daniell’s element in having nitric acid with a platinum electrode in the porous cell, instead of the copper solution and the copper electrode of Daniell's element. The hydrogen evolved at the platinum is oxidized by the nitric acid, and the polarization thus avoided. The nitrous fumes given off by the chemical action are very disagreeable, and also very poisonous, so that it is advisable to place the battery outside the experimenting room or in a suitable draught chamber. The electromotive force of Grove’s cell is a good deal higher than that of Daniell’s, and its internal resistance is very much less, , 25 ohm being easily attained with a cell of moderate dimensions. On this account the cell is much used for working induction coils, generating the electric light, and so on, notwithstanding that it is troublesome to fit up, and must be renewed every day.

In Bunsen’s element the platinum foils of Grove are replaced by carbon. The prime cost of the battery is thus considerably reduced, the more so now that carbons for the purpose have become articles of commerce. The electromotive force of the element thus altered is as great as, or with good carbons even greater than, in Grove’s original form; but the internal resistance is greater. There is a difficulty sometimes in obtaining good connection with the carbons, and trouble arises from their fouling; but the fact that this cell is a universal favourite in Germany proves its practical utility. It is comparatively little used in this country.

In the cell of Marié Davy, which is, or was, much used for telegraphic purposes in France, the copper solution and copper plate of Daniell are replaced by a wateιy paste of protosulphate of mercury, into which is inserted a carbon electrode. The electromotive force of this cell is said to be about 1 - 5 volts, and its internal resistance to be greater than that of Daniell’s cell.

Besides these, various bichromate elements of merit might be described; but we have dwelt long enough on this subject already.

The following table of Latimer Clark’s, quoted by Maxwell, will give the reader an idea of the relations as to electromotive force of the commoner elements :— [table]

The electromotive force is stated in volts, and the solutions in the third column are concentrated, unless it is otherwise stated.

Thermoelectricity.—We have already alluded to the law of Volta, according to which there can be no resultant electromotive force in a circuit composed solely of different metals; and it will be remembered that we added the condition that all the junctions must be at the same temperature. Seebeck was the first to discover that this law is subject to exception when the junctions are not all at the same temperature. If we form a circuit with an iron wire and a copper wire, and raise the temperature of one of the junctions a little above that of the other, a current flows round the circuit, passing from copper to iron over the hotter junction; similarly, if we solder together a piece of bismuth and a piece of antimony, and connect the free ends with the copper wires of a galvanometer, then when the junction of the bismuth and antimony is heated the galvanometer indicates a current passing from bismuth to antimony over the hot junction. It will be perceived that the second of our two illustrative instances is more complicated than the first, inasmuch as three metals enter into the circuit instead of two. Nevertheless the experimental result is not altered by the intervention of the copper wire (abstraction being made of its resistance), provided the temperatures of the points where it joins the bismuth and antimony respectively be the same. It is easy to give a direct experimental proof of this assertion by inserting between the pieces of bismuth and antimony a piece of copper wire so that the circuit now is Bi.Cu.Sb.Cu.Bi.; if the junctions of the inserted wire with the bismuth and antimony be raised to the same temperature as the BiSb junction in our second experiment, and the junctions with the copper wire of the galvanometer be at the same lower temperature as before, the total electromotive force in the circuit will be the same; and, provided the resistance of the circuit has not been sensibly increased by the interpolation of the copper wire, the galvanometer indication will also be the same as before The same result is obtained however many different metals we insert between the bismuth and antimony, provided the temperatures of all the junctions be the same and equal to that of the BiSb junction in the original experiment.

The law of Volta therefore still holds if stated thus: A series of metals whose junctions are all at the same temperature may be replaced by the two end metals of the senes without altervng the electromotive force in any circuit of which the series forms a part.

It is not unlikely that the above statement of the fundamental facts concerning thermoelectromotive force has suggested to the reader two notions:—1st, that the phenomena maybe completely explained by a contact force ai the junctions of the metals which is a function of the temperature of the junction; and 2d, that this contact force is the true contact force of Volta. It is perhaps as well to mention even at this early stage that the first of these notions is certainly not correct, and that the second is not admitted by some of the greatest authorities on the subject.

Seebeck examined the thermoelectric properties of a large number of metals, and formed a thermoelectric series, any metal in which is thermoelectrically related to any following one as bismuth (see above) is to antimony, the electromotive force in a circuit formed of the metals being ceteris paribus greater the farther apart they are in the series. The following is a selection from Seebeck’s series :—

Bi. Ni. Co. Pd. Pt. Cu. Mn. Hg. Pb. Sn. Au. Ag. Zn. Cd. Fe. Sb . Te.

This series has only a general interest, and is not to be rø garded as in any way absolute. Seebeck himself showed the great effect that slight impurities and variations of physical condition may have on the position of a metal in the series. Some specimens of platinum for instance come between zinc andcadmium. Another instance of the same kind is afforded by iron: Joule found the following order to hold—cast iron, copper, steel, smithy iron.

Thermoelectric series have been given by Hankel, Thomson, aud others, but we need not reproduce them here. It may be well, however, to direct the attention of the reader to the properties of metallic sulphides and of alloys which in many cases occupy extreme positions in the thermoelectric series. Alloys present anomalies in their thermoelectric properties somewhat similar to those already noticed in our discussion of their conductivity. These properties have been much studied with a view to practical applications in the construction of thermopiles. Considerable progress has been made in this direction (see above p. 11), notwithstanding the fact that many of the alloys most distinguished for their thermoelectric power are very brittle and have a tendency to instability under the continued action of heat.

Many measurements of the electromotive force of thermoelectric couples have been made by Matthiessen, Wiedemann, E. Becquerel, and others, but the results are of no great value owing to the effect of impurities and the want of sufficient data to determine all the thermoelectric constants of any one couple (see below, p. 99). Numerical data, such as they are, will be found in Wiedemann, Fleeming Jenkin’s Electricity and Magnetism, or Everett’s Illustrations of the Centremetre-gramme-second System oj Units. It will give the reader an idea of the order of the magnitudes involved to state that the electromotive force at ordinary temperatures of a BiSb couple is somewhere about 11700 C. G. S. absolute units when the difference between the temperatures of the junctions is 1° C. The corresponding number for a CuFe couple is 1600 or 1700.

Thermoelectric currents, or at least what may very likely be such, have been obtained in circuits other than purely metallic, e. g., in circuits containing junctions of metals and fluids, metals and melted salts, fluids and fluids. The phenomena in all these cases are complicated, and the results more or less doubtful; so that no useful purpose could be served by discussing the matter here. The same remark applies to the curious electrical phenomena of flames, of which no proper explanation, so far as we know, has as yet been given.

The experiments of Magnus have shown that in a circuit composed entirely of one metal, every part of which is in the same state as to hardness and strain, no thermoelectromotive force can exist, no matter what the variations of the section or form of the conductor or what the distribution of temperature in it may be (so long as there is neither discontinuity of form nor abrupt variation of temperature).

This statement is of great importance, as we shall see, in the theory of thermoelectricity. Its purport will be all the better understood if we dwell for a little on the three limitations which accompany it.

The great effect of the hardness or softness and crystalline or amorphous structure of a metal on its electric properties was observed by Seebeck soon after the discovery of thermoelectricity. The effect of temper in wires may be shown very neatly by the following experiment due to Magnus. On a reel formed by crossing two pieces of wood are wound several turns of hard-drawn brass wire softened in a number of places adjacent to each other on the reel. The free ends of the wire being connected with a galvanometer, and the parts of the wire lying between neighbouring hard and soft portions being heated, a thermoelectric curreut of considerable strength is obtained, whose direction is from soft parts to hard across the heated boundaries. Effects of a similar kind were obtained with silver, steel, cadmium, copper, gold, and platinum. In German silver, zinc, tin, and iron, the current went from hard to soft across the hotter boundary.

Sir V∖ ιlliam Thomson made a number of experiments on the effect of strain on the electric properties of metals. The results, some of them very surprising, are contained in his Bakerian Lecture, along with many other things of great importance for the student of thermoelectricity.

Two of his experiments may be described as specimens.

They afford convenient lecture-room illustrations of the subject under discussion. (1.) A series of copper wires A, B, C, D, E, F, G, <fcc., are suspended from a horizontal peg. A and B, C and D, E and F, &c., are connected by short horizontal pieces of copper wire, all lying in the same horizontal line, and B and C, D and E, F and G, <fcc., are connected by a series of pieces lying in another horizontal line below the former. An arrangement is made by means of which the alternate wires A, C, E, G, can be more or less powerfully stretched, while B, D, F, àc., are comparatively free. A piece of hot glass is applied to heat either the upper or lower line of junctions. A thermoelectric current is then observed passing from the stretched to the unstretched copper across the hot junctions. This thermoelectric current increases with the traction up to the breaking point. But the most remarkable point that comes out in such experiments is that when we free the wire after powerful traction, leaving it with a permanent set, there is still a thermoelectric current; but the direction is now from the soft or unstrained towards the permanently strained parts across the hot region. (2.) Iron gives similar results, only the direction of the current is in each case opposite to that in the corresponding case for copper. The following experiment exhibits this in a very elegant manner. One end of a piece of carefully annealed iron wire is wound several times round a horizontal peg, the free end being slightly stretched by a small weight, and connected with one terminal of a galvanometer. The other end of the wire is wound a few times round one side of a rectangular wooden frame, the free end being stretched by a small weight and connected with the other terminal of the galvanometer. The parts of the wire on the peg or the part on the frame is then heated, and weights are hung to the frame. As the weight increases, the deflection of the galvanometer goes on increasing. I f we stop a little short of rupture, and gradually decrease the weight, the deflection of the galvanometer gradually decreases to zero, changes sign before the weight is entirely removed, and finally remains at a considerable negative value when the wire is again free.

These experiments of Sir Wm. Thomson’s were repeated by Le Roux. The results of the two experimenters are not very concordant. This may be due to differences in the qualities of the materials with which they worked, or to the fact that Le Roux worked at higher mean temperatures than Thomson.

Le Roux also repeated the experiments of Magnus, confirming his general result, but adding the two last qualifications given above. He found, contrary to the result of Magnus, that when a lateral notch is filed in a . wire and one side heated, there is in general a thermoelectric current, which is greater, up to a certain limit, the deeper the notch. He also found that when two wires of the same metal, with flat ends, are pressed together, so that one forms the continuation of the other, and the wiιe on one side of the junction is heated, no current is obtained; but he observed a current in all cases where there was dissymmetry,—e.g., where an edge of one end was pressed on the flat surface of the other, where the wires overlapped or crossed, or where the chisel-shaped end of one wire fitted into a notch in the end of the other, and the axes of the wires were inclined, and so on.

Whether a very abrupt variation in temperature in a continuous part of a metallic wire would produce a thermoelectromotive force is a question which possesses little physical interest, since it is impossible to realize the imagined conditions. There can be no doubt, however, that, when the two unequally heated ends of a wire composed of the same metal throughout are brought together, a thermoelectric current is in general the consequence. Such currents were, it appears, observed by Ritter in 1801, when cold and hot pieces of zinc wire were brought into contact. Becquerel, Matteucci, Magnus, and others have experimented on this subject. The results obtained are, no doubt, greatly influenced by the state as to oxidation, àc., of the surfaces of the metals experimented on, as has been pointed out by Franz and Gaugain. The experimental conditions are, in truth, very complicated, and a discussion of the matter would be out of place here. We may mention, however, that, at the instance of Professor Tait, Mr Durham made experiments on the transient current which arises when the unequally heated ends of a platinum wire are brought into contact. It was found that the first swing of a galvanometer of moderately long period was proportioual to the temperature difference and independent of the mean temperature through a considerable range.

Cumming, who experimented on thermoelectricity about the same time as Seebeck, and apparently independently, discovered the remarkable fact that the thermoelectric order of the metals is not the same for high temperatures as for low. He found that, when the temperature of the hot junction in a circuit of iron and copper, or iron and gold, is gradually raised, the electromotive force increases more and more slowly, reaches a maximum at a certain temperature T, then decreases to zero, and finally changes its direction. The higher the temperature of the colder junction, so long as it is less than T, the sooner the reversal of the electromotive force is obtained. If the temperature of the hot junction be T + τ, where τ is small, then the reversal of the electromotive force takes place when the temperature of the colder junction is T - τ. If both junctione, A and B, be at the temperature T, then either heating or cooling A will cause a current in the same direction round the circuit, and either heating or cooling B will cause a current in the opposite direction.

The reversal of the current may be shown very conveniently in the manner recommended by Sir Wm. Thomson.

A circuit is formed by soldering an iron wire to the copper terminal wires of a galvanometer. If one junction be at the temperature of the room and the other at 300° C. or thereby, a current flows from copper to iron across the hotter junction; but, if we raise the temperature of both junctions over300°C., one being still a little hotter than the other (which can be managed by keeping both in a lamp flame, one in a slightly hotter place than the other), then the current will flow from iron to copper across the hot junction. If both junctions be allowed to cool, the difference between their temperatures remaining the same, the current will decrease, becoming zero when the mean temperature of the two junctions is about 280° C; and, on still further lowering the mean temperature, it will set again in the opposite direction, i.e., from copper to iron across the hot junction. The fundamental facts of thermoelectric inversion were confirmed by Becquerel, Hankel, Svanberg, &c.; but the matter rested there till it was taken up by Sir Wm. Thomson in the course of his classical researches on the applications of the laws of thermodynamics to physical problems.

The application of the first law of thermodynamics leads to no difficulty; and it indicates that the heat absorbed according to Peltier’s law, in the ordinary case when a current passes from copper to iron across the hotter of the junctions, minus the heat evolved at the colder junction where the current passes from iron to copper, is to be looked on as a source of part at least of the energy of the thermoelectric current. If absorption or evolution of heat occur anywhere else than at the junctions, this must be taken account of in a similar manner.

The application of the second law is of a more hypothetical character. It is true that the Peltier effects, as we may for shortness call the heat absorption and evolution at the junctions, are reversible in this sense that we might suppose the thermoelectric current, whose energy arises wholly or partly from the excess of the heat absorbed at the junction A over that evolved at the junction B, used to drive an electromagnetic engine and raise a weight; and that we might suppose the potential energy thus obtained again expended in sending, by means of an electromagnetic machine, a current in the opposite direction round the circuit, absorbing heat at B, evolving heat at A, and thus restoring the inequality of temperature. This process, however, must always be accompanied by dissipation of energy, (1) by the evolution of heat in the circuit according to Joule’s law, and (2) by conduction from the hotter towards the colder parts of the wires. The first of these effects varies as the square of the current strength, while Peltier’s effect varies as the current strength simply; so that the former might be made as small a fraction of the latter as we please by sufficiently reducing the current, and thus, theoretically speaking, eliminated. The second form of dissipation could not be thus got rid of, and could only be eliminated in a circuit of infinitely small thermal but finite electric conductivity, a kind of circuit not to be realized, as we know (see above p. 51). Still it seems a reasonable hypothesis to assume that the Peltier effects, and other heat effects if any, which vary as the first power of the strength of the current, taken by themselves are subject to the second law of thermodynamics. Let us now further assume that all the reversible heat effects occur solely at the junctions. Let II, II' denote the heat (measured in dynamical equivalents) absorbed and evolved, at the hot and cold junctions respectively in a unit of time by a unit current. Let E be the electromotive force of an electromotor maintaining a current I, in such a direction as to cause absorption of heat at the hot junction. Then, if R be the whole resistance of the circuit, we have, by Joule’s law and the first law of thermodynamics, EI + ∏I-Π'1 = RI 2 , (1), supposing the whole of the energy of the current wasted in heat. Hence we get i . e + ∏- π (2) .

It appears then that, owing to the excess of the absorption of heat at the hot junction over the evolution at the cold iunction, there arises an electromotive force ∏ - IT helping to drive the current in the direction giving heat absorption at the hot junction. We may suppose (and shall henceforth suppose) that E = 0, and then the current will be maintained entirely by the thermoelectromotive force.

If we now apply the second law, we get

5!-ι!-o, θ θ ’ «

θ and θ' being the absolute temperatures of the hot and cold junctions. Hence π -∏' . . . 3); e θ'

or, in other words, ∏ = C0, where C is a constant depending only on the nature of the metals. In accordance with this, the thermoelectromotive force in the circuit would be C(0 - O'); that is, it would be proportional to the difference between the temperatures of the junctions. Now this conclusion is wholly inconsistent with the existence of thermoelectric inversions. We must therefore either deny the applicability of the second law, or else seek for reversible heat effects other than those of Peltier. This line of reasoning, taken in connection with another somewhat more difficult, satisfied Sir Win. Thomson that reversible heating effects do exist in the circuit elsewhere than at the junctions. These can only exist where the current passes from hotter to colder parts of the same wire or the reverse. Thomson was thus led to one of the most astonishing of all his brilliant discoveries; for he found, after a series of researches distinguished alike for patience and experimental skill, that an electric current absorbs heat in a copper conductor when it passes from cold to hot, and evolves heat in iron under similar circumstances. This phenomenon was called by its discoverer the electric convection of heat. He expressed the facts above stated by saying that positive electricity carries heat with it in an unequally heated copper conductor, and negative electricity carries heat with it in an unequally heated iron conductor. The first statement is perhaps clearer; the value of the one given by Thomson consists in the suggestion which it conveys of a valuable physical analogy with the transport of heat by a current of water in an unequally heated pipe.

If two points AB of a uniform linear conductor, in which a current I is flowing from A to B, and evolving heat, be kept at the same constant temperature, but for the electric transport of heat the temperature distribution would be symmetrical about a point of maximum temperature half way between A and B Owing to the electric transport of heat, the maximum will be shifted towards A in iron, towards B in copper. This remark contains the principle of the experiments made by Thomson to detect the new effect.

The first experiment in which the effect was satisfactorily established was made with a conductor ABCDEFG, formed of a number of strips of iron bound together at A, C, E, and G, but opened out widely at B, D, and F, to allow these parts to be thoroughly heated or cooled. At C and E small cylindrical openings allowed the bulbs of two delicate mercurial thermometers to be inserted in the heart of the bundle of strips. The part D of the conductor was kept at 100° C. by means of boiling water, and the parts B and F were kept cool by a constant stream of cold water. The current from a few cells of large surface was sent for a certain time from A to G, then for the same length of time from G to A, and so on. In this way the effects of want of symmetry were eliminated, and the result was that the excess of the temperature at E over that at C was always greatest when the current passed from G to A; whence it follows, as stated above, that a current of positive electricity evolves heat in an iron conductor when it passes from cold to hot.

Le Roux has made a series of interesting experiments on the Thomson effect in different metals. He found that the effect varies as the strength of the current, and gives the following numbers representing its relative magnitudes in different metals. In lead the effect is insensible. [table]

We may now apply the mathematical reasoning given above, taking into account Thomson’s effect.

Suppose for simplicity we have a circuit of two metals only. Let the current go from A to B over the hot junction, and let the heat absorbed in passing from a point at temperature 0 to a neighbouring point at temperature θ + dθ in A be σ λ dθ per unit of current per unit of time; let σ. 2 dθ be the corresponding expression for B. Then it is obvious, from the result of Magnus (see above, p. 95), that σ 1 and σ i can be functions of the temperature merely; they depend, of course, on the nature of the metal, but are independent of the form or magnitude of the section of the conductor. The first and second laws now give respectively

E = Π-Π'+∕∖ 0 (<r 1 -σ 4 )rf0 .... (4),

∕ 0

Π ∏ ∕^θ a ’ ι (r2 j λ ∕g∖

· < 5 >∙

where E is the whole thennoelectromotive force, and ∏ and ∏' are the same functions of θ and θ ’ respectively. By differentiation we get from (5)

a(i)÷V=∙ · · · · < ’ >≈

whence we easily get e √>)

- z ^ β U . . . . (7>.

or 5=⅛ (

θ dθ )

This last equation enables us to determine E in terms of ∏, and conversely.

When the difference between the temperatures of the junctions is very small, equal to dθ say, the thermoelectromotive force is "do (8).

The coefficient by which we must multiply the small temperature difference to get the electromotive force is called by Thomson the thermoelectric power of the circuit. If we have a circuit of three metals, A, B, C, all at the same temperature 0, then we know that

∏BC ÷ ∏CA ÷ ∏AB = 0, whence ∏λb ∏λc _ ∏bc . q . .

θ “θ θ ’

or, in other words, the thermoelectric power of B with respect to A is equal to the difference between the thermoelectric powers of a third metal C with respect to A and B respectively.

Thus far we have been following Thomson. But as yet we have no indication how σ, the coefficient of the Thomson effect, depends on the temperature. Thomson himself seems (see his Bakerian Lecture, l. c., p. 706) to have expected that σ would turu out to be constant. Certain considerations concerning the dissipation of energy led Tait, however, to conjecture that σ is proportional to the absolute temperature. If we adopt this conjecture, Thomson’s equations give us at once the values of the Peltier effect and the electromotive force in the circuit. If σ 1 = kf), σ 2 = k 2 θ, we get from (6) and (7) successively

Π = (⅛ 1 -i i )(β li -0)0 (10),

E=(⅛ 1 -⅛,)(β-r){0 lt -4(∙+β')} · · . (∏), where 0 l2 is the neutral temperature. Also, since in a circuit of uniform temperature there are no Thomson effects, and the sum of the Peltier effects is zero, we get for any three metals

(k t — i' 3 )0 2 J + (k 3 — i'ι)θ3i + (kχ — kj)θ i , = 0 . . (12).

Taking up the idea of a thermoelectric diagram originally suggested by Thomson, Tait has shown how to represent the above results in a very elegant and simple manner. Suppose we construct a curve whose abscissa is the absolute temperature 0, and whose ordinate is the thermoelectric power of some standard metal with respect to the metal we are considering, then, from what has been shown (10), Tait’s conjecture leads to the result that this curve is a straight line; and if the standard metal be lead, for which, according to Le Roux’s results, the Thomson effect is zero, then the coefficient k of the Thomson effect is the tangent of the inclination of the representative line to the axis of abscissae. And not only so, but it follows from formulae (9) and (7) that, if A ’ AN, B'BN (fig. 54) be the lines corresponding to two metals, say Cu and Fe (of which the former is above the latter in the thermoelectric series at ordinary temperatures), and if AB, A ’ B' be the ordinates corresponding to θ and θ', then the electromotive force in a circuit of the two metals whose junctions are at the temperatures θ and 0', tending to send a current from Cu to Fe across the hotter junction, is represented by the area ABB'A'. The Peltier effects at the two junctions are represented by the rectangles ΑΒάα and A'B'6'α', and the Thomson effects, in the Cu and Fe respectively, by AA'DC and BB'GF, or by ΑΑ'ό and BB'6'δ, which are equal to these. At N, where the lines intersect, the Peltier effect vanishes. N therefore is the neutral point; and, if the higher temperature lie beyond it, the electromotive force must be found by taking the difference of the areas NA'B' and NAB, and so on. All the phenomena of inversion may be studied by means of this diagram, and the reader will find it by far the best means for fixing the facts in his memory.

For several years back Tait and his pupils have been engaged in verifying the consequences of this conjecture: and it has been shown, first, for temperatures within the range of mercury thermometers, and latterly for temperatures considerably beyond this range, that the hypothesis accords with experience. The methods employed by Tait in his experiments at high temperatures are of great interest and importance. One of these was to construct a curve whose ordinate and abscissa are the simultaneous readings of two thermoelectric circuits whose hot and whose cold junctions are kept at common temperatures. It is a consequence of the foregoing assumption that the curve thus obtained ought to be a parabola. Very good parabolas were in many cases obtained. In some cases, however, the curves, so far from being parabolas, were actually curves having points of contrary flexure. This anomaly led Tait to the discovery of the astonishing fact that the Thomson effect in iron changes its sign certainly once at a temperature near low red heat, if not a second time near the melting point. It was found that the inflected curves could be represented by piecing together different parabolas. Hence the line for iron in the thermoelectric diagram is a broken line made up of two if not three straight pieces. This peculiarity of the iron line was very strikingly shown by forming circuits of iron with the alloys Ptlr or PtCu. Such circuits exhibit two or even three neutral points (see fig. 55). Another very elegant method of verification consisted in using along with an iron wire a multiple wire of Au and Pd, the resistances of whose branches could be modified at will. It is easy enough to show that the line for the Au-Pd wire is a straight line, passing through the neutral point of Au and Pd, and such that it divides the part of an ordinate lying between the Au and Pd lines in the ratio of the respective conductivities of the Au and Pd branches. Thus, by increasing ratios of the conductivities of the Pd and Au branches from 0 up to ∞ , we can make the Au-Pd line sweep through the whole of the space between Au and Pd (fig. 55), and thus explore the part of the Fe line lying in the space. We get in this way first one neutral point, then two, then one, and then none in our Fe, Au-Pd circuit.

Tait has pointed out that, by using Ptlr and Fe, and keeping the hot and cold junctions at the two neutral temperatures, we get a current maintained solely by the excess of the heat absorbed in the hotter iron over that developed in the colder. The electromotive force is represented by the area inclosed by the part of the zigzag on the Fe line cut off by the Ptlr line (fig. 55). A similar case of thermoelectromotive force without Peltier effects may be obtained with three metals, such as Fe, Cd, Cu, whose neutral points lie within reasonable limits. The electromotive force in this case is represented by the triangle between the three lines.

We subjoin a table, calculated by Professor Everett from Tait’s diagram. The thermoelectric power is given in electromagnetic (C.G.S.) units, in terms of the temperature (<) in centigrade degrees, by means of the formula α + βt, where a and β have the tabulated values :— [table]

We need scarcely warn the reader that the results in this table must not be rashly applied to any specimens of the metals taken at random. The temperature limits lie between 18° C. and 420° C.

It would be extremely interesting to compare the results of absolute measurements of the Peltier effect with Tait’s theory; but, unfortunately, no data that λve know of are available for the purpose. It is absolutely necessary for this purpose to have heat measurements and determinations of the lines of the metals in the same specimens. The data of Edlund and Le Roux are quite useless for such a purpose. One result of Le Roux’s is, however, interesting. He finds for the amount of heat developed at the junction BiCu, the values 3Ό9 and 3 - 95 at 25° C. and 100° C. respectively. Since the neutral temperature of BiCu is very high, the Peltier effect ought, according to Tait’s theory, to vary as the absolute temperature. The absolute temperatures corresponding to 25° C and 100° C. are 298° and 373°, and we have 3 - 95 ÷3O9 = D278, while 373 ÷ 298 = 1’252; the agreement between these numbers bears out the theory so far.

General Considerations regarding the Seat of Electromotive Force.—Before proceeding to notice the remaining cases of the origin of electromotive force, in which the phenomena are more complicated, and the experimental conditions less understood, it may be well to call attention to a principle that appears to hold in most of the cases already examined. In most of these cases the seat of the electromotive force appears to be at the places where energy is either taken in or given out in the circuit.

It is very natural to ask ourselves what the consequences would be if we applied this principle to the voltaic circuit. It would probably be admitted by most that the energy in the voltaic circuit is taken in mainly at the surface of the electropositive metal. This admission, taken in conjunction with the general principle above stated, leads us to the conclusion that the electromotive force resides mainly at the surface of the electropositive metal. The absorption or evolution of energy at the junction of the dissimilar metals is quite insignificant, and we should, on the same view, deny that any considerable part of the electromotive force resides there.

This view appears to be at variance with the theory of metallic contact, as now held by Sir William Thomson and others; and the burden of explaining the experiments made by him and others on the contact force of Volta is doubtless thrown on those who adopt this view. The position of such would very likely be that there is an uneliminated source of uncertainty in all these experiments (see above, p. 85). On the other hand, those who adopt the contact force of Volta at the junction of copper and zinc as the main part of the electromotive force of Daniell’s element are under the necessity of distinguishing this from the electromotive force corresponding to the Peltier effect, which must be a distinct effect, since it is but a very small fraction of that of a Daniell’s cell.

We are, however, so very ignorant of the nature of the motion which is the essence of the electric current that the very form in which we have put the question may be misleading. If this motion be in the surrounding medium, as there is great reason to believe it to be, it would not be surprising to find that speculations as to the exact locality of the electromotive force in the circuit were utterly wide of the mark. The very language which we use implies a certain mode of analysing the problem which may be altogether wrong. The only thing of which we can as yet be sure is that the mathematical equations deduced from Ohm’s law and other proximate principles are in exact accordance with experiment.

Pyroelectricity.—Some account of this interesting subject has already been given in the Historical Sketch at the beginning of this article. It will be well, however, to state here some of the conclusions of those who have recently investigated the matter. It seems now to be settled that it is not merely high or low temperature, but change of temperature, which gives rise to the electrical phenomena of pyroelectric crystals. The properties exhibited by tourmaline may be described thus. One end A of the crystal is distinguishable from the other end B by the dissymmetry of the crystalline form. A is called the analogous pole of the crystal, and B the antilogous pole. When the temperature of the crystal is increasing uniformly throughout, the analagous pole is positively electrified and the antilogous pole negatively electrified. When the temperature is decreasing uniformly throughout, the analogous pole is negative and the antilogous pole positive. This law was originally discovered by Canton, but it seems to have been lost sight of again and rediscovered both by Bergman and by Wilcke in 1766. When the temperature is uniform, the positive and negative regions are symmetrically distributed about the central zone of the crystal, which is neutral. If the ends be unequally heated, this symmetry no longer obtains. It must not be forgotten that complications may arise from the crystal becoming electrical as a whole by friction, usually positive, like most other vitreous bodies.

Gaugain made a series of interesting experiments on the electrical properties of tourmaline, and concluded that a tourmaline whose temperature is varying may be compared to a voltaic battery of great internal resistance, consisting of an infinite number of cells, each of infinitely small electromotive force; so that the electromotive force is proportional to the length of the tourmaline, and its internal resistances is proportional to the section inversely and to the length directly. He also concluded that the amount of electricity furnished by a tourmaline, w hile its temperature varies either way between two given temperatures, is always the same.

In order to explain the properties of the tourmaline, it has been supposed that the crystal is naturally in a state of electrical polarization, like that assumed by Maxwell in a medium" under the influence of electromotive force, or more nearly (since no sustaining force having an external origin is supposed) like that of a permanent magnet. The intensity of this polarization is supposed to be a function of the temperature. Supposing the tourmaline to remain for some time at the same temperature, a surface layer of electricity would be formed, which would completely mask the electrical polarization of the crystal, inasmuch as it would destroy all external electrical action. This neutralization would be instantly effected by running the cιystal through the flame of a lamp. If, however, the temperature increase, then the polarization will, let us say, increase, so that the surface electrification no longer balances it. We shall thus get polar electrical properties of a certain kind. If the temperature decrease, the polarization will decrease, and we shall thus get polar properties of the opposite kind.

In many pyroelectric crystals there are more than one electric axis, so that we have several analogous and corresponding antilogous poles. An enumeration of the various crystals in which pyroelectric properties have been found, and a discussion of the peculiarities in their crystalline form, belongs more properly to the science of Mineralogy. Much has been done in this department by Köhler, Gustav Rose and Riess, and Hankel. For some very interesting researches by Friedel see Annales de Chimie et de Physique, 1869.

Frictional Electricity.—In accordance with the general principle laid down at the beginning of this section, we should expect to find an electromotive force at the surface which separates two different non-conducting media, just as we have found it at the boundary of two different conducting media. The effect of such a contact force would be very different however in the former of these cases, from what we have seen it to be in the latter. In the case of non-conductors the electricity cannot leave the surface of separation, but will simply accumulate on the two sides of it, till the force arising from electrical separation is equal to the contact force. On separating the bodies, in certain cases, we may carry away with us these surface layers of electricity, and it is an obvious consequence of our principles that the electrifications of parts of the two bodies that have been in contact must be equal and opposite. While the bodies are in contact the difference of potential between the layers of electricity corresponding to very considerable surface density may be very small, just as in Volta’s condensing electroscope (see above, p. 34); but when we separate the bodies work is done against the electrical attractions, and the potential increases enormously.

These hypothetical results tally very well with the electrical phenomena observed when non-conducting bodies are lightly rubbed together; and the above is nearly the explanation that most physicists of the present day would probably give (if they gave any) of what is called the “frictional generation of electricity.”

All experimenters are agreed that equal quantities of positive and negative electricity appear in this case as in every other case of electrical separation; an experiment to prove the contrary would have to be very demonstrative indeed before it would now be accepted as conclusive. A single case of exception would revolutionize our fundamental ideas completely. The reader should consult on this point Faraday’s Experimental Researches, series xi. 11 ii.

The other consequences of our hypothesis are by no means so firmly established. One of these is that we ought to be able to arrange non-conducting bodies in a series such that any body rubbed with one below it in the series becomes positive, and rubbed by one above it negative.

Many electricians have attempted to establish such electromotive series, but the experimental conditions (see the admirable remarks of Riess, Reibibngseleetrieität, § 907) are so complicated that nothing absolute has been attained. Yet it would appear that, if we could make sure that we were always dealing with definite materials under definite surface conditions, electromotive series could be constructed in which every different body would have a fixed position. As it is, the body bearing the same name in the lists of different experimenters was in all probability not exactly of the same material in all cases, and (we might say certainly) was not under the same surface conditions. We refer the reader to Riess (Z.c.) for an admirable résumé of the work of different electricians in this department. Mascart has given a very interesting account of the matter (t. ii. § 834, &c.) from a more modern point of view. From these sources, together with indications in Young’s Lectures on Natural Philosophy, the reader will be able to follow up the literature of this somewhat uninviting department of electricity.

We give two instances of frictional electromotive series which mav be useful in giving the reader a general idea how different bodies stand.

The following is Wilcke’s series (1758) :—Glass, woollen cloth, feathers, wood, paper, shellac, white wax, ground glass, lead, sulphur, metals.

Faraday gives—cat and bear skin, flannel, ivory, feathers, rock crystal, flint glass, cotton, linen, white silk, the hand, wood, shellac, metals (iron, copper, brass, tin, silver, and platinum), sulphur.

To which Riess adds (in order) the highly negative bodies—guttapercha, electrical paper, collodion, gun cotton.

Considered as evidence for the contact hypothesis, the experiments of Peelet seem to be important. He used an apparatus which was virtually a Nairne’s machine (see below, p. 101), in which the rubber could be varied at will. His general conclusions are quite in accordance with the contact' theory. He found, for instance, that for the great majority of materials the quantity of electricity generated was independent of the pressure and of the breadth of the rubber, and varied as the angular velocity of the cylinder, and it even appeared to be the same for rolling friction as for sliding friction, so long as the material of the rubber was unchanged.

Besides the case of two non-conductors, we might consider the case of a conductor and a non-conductor in contact. Much of what has just been said would apply to this case also, an excellent . example of which is furnished by a frictional electrical machine of the ordinary construction when the cushions are well furnished with amalgam. This is the place to give a short account of these time-honoured pieces of electrical apparatus. For fVhistory of . them we cannot do better than refer to Mascart (Z.c.), who has devoted much attention to the theory as well as the history of electrical machines in general.

A very common form of machine, called Ramsden’s, is pictured in fig. 56. It consists, like all other frictional machines, essentially of three parts—(1) the rubbed or moving body, (2) the rubbers, and (3) the collectors and prime conductors. In the present instance the rubbed body is a disc of glass, which can be turned about a horizontal axis by means of a suitable handle. The efficiency of the machine depends very much on the quality of the glass of which the disc is made. According to Mascart, glass of old manufacture is superior to the more modern specimens, owing to the smaller proportion of alkali in the former; it appears, however, that the disc improves in most cases with age and use. Many other materials have been proposed to replace glass, which is somewhat costly when large discs are required. Ebonite has been tried a good deal of late, and has great advantages so far as its electrical properties are concerned; but it has the disadvantage that it warps very readily if heated incautiously, and its surface will not keep good for any length of time. Owing to decomposition under the action of light, a layer of sulphuric acid forms on the surface, after which it is very difficult to restore the electrical virtue so remarkable in the new material, although washing with hot water or immersion in a blast of steam are said to be effective in some degree.

The rubbers consist of two rectangular pieces of wood, hinged to supports attached to the framework of the machine, and fitted with springs and screws, so that they can be made to clip the plate with any required pressure. The rubbing surfaces are usually formed of leather, stretched as smooth and flat as possible (oiled silk' is sometimes used, but it is not so durable). Before the leather cushions are fit for use, they must be carefully coated with amalgam. The amalgam most commonly used is Kienmayer’s, which is a composition of two parts of mercury with one of zinc and one of tin. A great variety of different compounds of this kind have been used by different electricians, bisulphide of tin being a general favourite. The amalgam must be powdered as finely as possible, all grit being carefully removed. The cushions are then to be lightly smeared with lard, and worked together till the surface is very smooth and the greasiness almost gone; then the amalgam is to be carefully spread over them, and the surfaces again worked together till a uniform metallic surface is attained; they are then ready for use. The amalgam aids the action of the machine in two ways,—first, by presenting a surface which is highly negative to glass; secondly, by allowing the negative electricity evolved by friction to flow away without hindrance from the points of contact. In order to secure the second of these advantages still more perfectly, the cushions should be carefully connected by strips of tinfoil, or otherwise, with knobs, which can be put to earth during the action of the machine.

The collectors are two stout metal forks bestriding the glass disc at the ends of a horizontal diameter. They are armed, on the sides next the glass, with rows of sharp points, which extend across the rubbed part of the disc. The prime conductor in the specimen we are describing forms a metal arch rising over the framework of the machine, and insulated from the sole by two glass pillars. Various forms are given to this part of the machine, according to the fancy or convenience of the experimenter. One important thing to be seen to is, that there be no salient points on it which might facilitate the dissipation of electricity by brush, convective, or spark discharge.

After what has been said, the action of the machine requires little explanation. The disc, electrified positively by contact with the amalgam, carries away a positive charge, whose potential rises rapidly as it leaves the cushion,—so high, in fact, that there is a tendency to discharge to the air, which is prevented by covering the excited parts of the disc by pieces of oiled silk. When the highly charged glass comes opposite the points of the collector, owing to the inductive action, negative electricity issues from the points and neutralizes the charged plate, which at this point is virtually inside a closed conductor. The result of this is that the prime conductor becomes positively charged. The neutralized parts of the disc now pass on to be rubbed by the other cushion, and so on. The electricity goes on accumulating in the prime conductor until the potential is so great that discharge by surface conduction, or by spark, takes place between the collectors and the cushion, or between the collectors and the axis.

If it is desired to obtain negative electricity from a machine with a glass disc, we have simply to connect the prime conductor to earth, insulate the cushions, and collect the electricity from them.

We have said that there is a limit to the potential to which tho charge on the prime conductor can be raised. V e can never get a longer spark from the machine than the length of the interval between the collector and the cushion or the axis, as the case may be. The limiting potential can, however, be increased by insulating the axis of the machine, or making the axis itself wholly or partially of insulating material, and by using only one rubber and one collector, and placing them at the extremities of a diameter. The machine of Le Roy, often called Winter’s machine (fig. 57), is constructed on this pattern. We get, of course, caderis paribus, only half as much electricity per revolution with a machine of this kind as with Ramsden’s; but the spark is longer, in consequence of the greater insulation between the cushion (A) and the collector (B).

The cylinder machine, also called Nairne’s machine, was one of the first machines in which all the essential parts of the modern frictional machine appeared. It consists of a glass cylinder, which can be turned about a horizontal axis by a multiplying gear, or (as is now more usual) by means of a winch handle simply. The cushion is affixed to one horizontal metal cylinder, and the collector to another. It is necessary to insulate the axis in this machine, owing to its proximity to the ends of the conductors. Positive or negative electricity can be obtained with equal readiness by insulating either of the conductors, and connecting the other with the earth.

Those who desire more minute information concerning the functions of the different organs of the frictional machine, are referred to Mascart, tom. ii. § 834, &c. In the same place will be found a description of the famous machine with double plates constructed by Cuthbertson for Van Marum, and still to be seen in Teyler’s Museum at Haarlem. A description of another of Van Alarum's machineswill be found in the article “Electricity” in the Encyclopaedia Metropolitana. We take this opportunity of calling the scientific reader’s attention to that article, which contains a great quantity of very valuable matter. Much of the work of the earlier electricians that we have been obliged to pass over in silence is fully described there.

Electric machines have also been constructed of less costly materials than glass or even vulcanite—of cloth and paper, for instance —for an account of these, see Riess, Rd. ii. §§ 936, 937.

Many experiments have been made on the electrification of sifted powders. We have already, in describing Lichtenberg’s figures, alluded to some cases of this kind. As a rule, either the results are very uncertain, or the conditions of the experiment very complicated, so that the experiments are, in most cases, more curious than valuable, from a scientific point of view. Such as desire it will find abundant indications of the sources of information in Riess, Bd. i. §§ 938 sqq., and Ency. Metrop., art. “Electricity,” §§ 193 sqq. One case of this kind, however, was so famous in its day, that we ought to mention it. In the year 1840 a workman at Newcastle, having accidentally put one hand in the steam which was blowing oil at the safety valve of a high-pressure engine boiler, while his other hand was on the lever of the valve, experienced a powerful electric shock in his arms. Armstrong investigated the matter, and was led to construct his famous hydroelectric machine. This apparatus consists simply of an insulated boiler for generating high-pressure steam, fitted with a series of nozzles, kept cool by a stream of water. The steam issues from these nozzles and impinges on a conductor armed with points for collecting the electricity. The boiler gets electrified to a high potential, and a torrent of dense sparks may be drawn from it. The machine far surpassed any ordinary electrical machine in the quantity of electricity furnished in a given time. By means of it water was decomposed, and the gases collected separately. It was reserved for Faraday to trace the exact source of the electromotive force. He demonstrated, by a series of ingenious experiments, that the electrical action arose from the friction of the particles of water in the condensed steam against the wood of the nozzles.

Remaining Cases.—Of these the most important are atmospheric electricity, which belongs properly to meteorology, animal electricity, comprehending the study of the properties of the electrical fishes, and the electric phenomena of nerve and muscle. We have already indicated the literature of the former subject, and the latter belongs, for the present at least, to physiology. Evaporation, combustion, and in fact chemical action generally, have been brought forward by some experimenters as sources of electromotive force. About the last of all there is, of course, in one well-known case no doubt. As to the experiments generally alluded to under the other two heads—in particular, those of Laplace and Lavoisier, Volta, Pouillet, and others—there has been considerable difference of opinion, and we need not occupy space here with fruitless discussion of the matter. Similar remarks apply to the electrification caused by pressure, cleavage, and rapture.

Machines founded on Induction and Convection. —The oldest electric machine on this principle is the electrophorus of Volta. 1775. This consists of a plate of resinous matter (now usually vulcanite) backed by a plate of metal, and a loose metal plate, which we may call the collector, fitted with an insulating handle. The vulcanite is electrified by flapping it with a cat-skin, the collector is placed upon it, uninsulated for a moment by touching it with the finger, and then lifted by the insulating handle. The collector plate is then found to be charged (positively) to a high potential, and sparks of some length may be drawn from it. The explanation of the action of the electrophorus is simple enough, if we keep clearly in view the experimental fact that the surface electrification of a non-conductor, like vulcanite, will not pass to a metal plate in contact with it under ordinary circumstances. If the surface density of the electrification be very great, discharge to the metal may no doubt take place; and if the collector be kept fur a very long time in contact with the vulcanite, it is said that it may become negatively electrified. In the normal state, however, the negative electricity of the vulcanite remains upon it, and the thin layer of air intervening between it and the collector forms the dielectric in a condenser of very great capacity, so that a quantity of electricity collects on the lower surface of the condenser very nearly equal to that on the vulcanite. The difference of potential between the plates is very small (just as in Volta's condensing electroscope, see above, p. 34). When the collector is raised it carries away the positive charge—the potential of which, owing to the decrease in the capacity of the collector, rises enormously. It is to be noticed that the potential of the charge on the vulcanite rises to a corresponding extent. This remark partly explains the remarkable fact that, when the collector is kept on the excited vulcanite, its electrification may be kept for a long time (for weeks under favourable circumstances), whereas it speedily dissipates if the vulcanite be left uncovered. According to Riess, the fact that a plate of metal laid on an excited piece of glass tends to preserve its electrification was discovered by Wilcke in 1762. )

If each time we charged the collector it were discharged by contact with the interior surface of a hollow conductor A, it is obvious that we could raise A by a sufficient number of such contacts to as high a potential as we please, provided it were sufficiently well insulated. This remark brings Volta’s electrophorus into the present category of electrical machines.

In the rest of the induction machines to be described the excited dielectric is dispensed with, and an electrified conductor substituted in its place.

The earliest apparatus that involved the principle of such machines appears to have been Bennet’s doubler. The principle of this apparatus may be explained thus. Let A and C be two fixed discs, and B a disc which can be brought at will within a very short distance of either A or C. Let us suppose all the plates to be equal, and let the capacities of A and (J in presence of B be each equal to p, and the coefficient of induction between A and B, or C and B, be q. Let us also suppose that the plates A and C are so distant from each other that there is no mutual influence, and that p! is the capacity of one of the discs when it stands alone. A small charge Q is communicated to A, and A is insulated, and B, uninsulated, is brought up to it; the charge on B will be - Jq. B is now uninsulated and brought to face C, which is uninsulated; the charge on C will be -,Q. C is now insulated and connected with A, which ρ- ’

is always insulated. B is then brought to face A and uninsulated, so that the charge on A becomes rQ, where

'w('<

A is now disconnected from C, and here the first operation ends. It is obvious that at the end of n such operations the charge on A will be r n Q, so that the charge goes on increasing in geometrical progression. If the distance between the discs could be made infinitely small each time, then the multiplier r would be 2, and the charge would be doubled each time. Hence the name of the apparatus.

Darwin, Cavallo, and Nicholson devised mechanism for effecting the movements which in Bennet’s instrument were made by hand. Cavallo’s was a reciprocating movement, but in the machines of Darwin and Nicholson the motion was continuous and rotatory. Nicholson’s doubler is a very elegant instrument. A drawing of it is given by Mascart (t. ii. § 845)]; the specimen there represented is very like one which was found among the late Professor Willis’s apparatus, and is now in the Cavendish Laboratory at Cambridge. A still more elegant machine is “Nicholson’s spinning condenser,” which bears a remarkable resemblance to the induction machine of Topler. A description, with a figure, will be found in the Encyclopaedia Metropolitana, art. “Electricity,” § 112.

It is obvious that if any conductor be connected with the part of any of these machines corresponding to the conductor A in the above description, and the potential of A be raised to any small positive or negative value, we can by means of the machine increase the charge, and therefore the potential, up t<5 any required amount. We have, in fact, an electric machine which may be used for all the ordinary purposes. It was not with this view, however, that these pieces of apparatus were first invented, but rather for the purpose of demonstrating small electric differences. In this they were but too successful, for it was found that it was impossible to prevent them from indicating electric differences unavoidably arising within the apparatus itself. It was this difficulty no doubt that led to their being ultimately abandoned, and for a time forgotten, although they were once in high favour. Of late, however, they have been taken up as electromotors with great success.

The type of all these machines is an arrangement of the following description. A conductor or carrier C, or a series of carriers, is fastened upon the circumference of an insulating disc. At the ends of a diameter are two hollow conductors, A and B, embracing the disc on both sides, so that twice in the course of a revolution the carrier is virtually in the interior of a hollow conductor. Inside each conductor are two springs: one of these is in metallic connection with the conductor, and may be called the receiving spring; the other, called the inductor spring, is insulated from the conductor, and is connected either to earth or with the corresponding spring belonging to the other conductor. Suppose A to be at a small positive potential, and B at zero potential; starting with C in connection with the inductor spring inside A, it becomes negatively electrified and carries away its charge; it next conies iιι contact with the receiving spring in B, and, being now part of the interior of a hollow conductor, it parts with the whole of its charge to B; then it passes on and is charged positively at B s inductor spring; then discharges to A at A’s receiving spring; and so on. 'Γhe positive and negative charges are each a little increased every revolution, and the difference of potentials accordingly augmented. This is the principle of Varley’s machine (i860), and of Thomson’s mouse mill and replenisher (1867); it is virtually that of Bennet’s doubler.

Closely allied to these machines is Thomson’s water-dropping potential equalizer. This consists of an insulated reservoir of water, with a long pipe, from the nozzle of which water is allowed to break in drops. It is obvious that if the potential of the reservoir be above that of the air surrounding the spot where the water breaks into drops, each drop will carry away with it a positive charge, and this will go on till the potentials are equalized. This device was introduced by Thomson in observations on atmospheric electricity. The burning match which he U3es in conjunction with the portable electrometer acts in the same way. He has also constructed a water-dropping electric machine on a similar principle. Two streams of water break into drops inside two inductors connected with the internal armatures of two Leyden jars, A and B; the drops from each inductor fall into a receiver connected with the other inductor. A very small difference of potential between the jars starts or reverses the action of the apparatus; in fact, it will in general start of itself, and very soon sparks are seen passing between the different parts, and the drops are scattered in all directions by the strong electrical forces developed.

The most remarkable, as well as the most useful, of all these machines is that of Holtz. Here the convection is effected by means of a disc of glass, which is mounted on a horizontal axis F (fig. 58), and can be made to rotate with considerable angular velocity by means of a multiplying gear, part of which is seen at X. Close behind this glass disc is fixed another vertical disc of glass, in which are cut two windows, B, B. On the side of the fixed disc next the rotating disc art pasted two sectors of paper, A, A, with short blunt points attached to them, which run out into the windows towards the rotating disc, without quite touching it. Two metal combs C are placed on the other side of the rotating disc (that nearest the reader), the teeth being put opposite the parts of A, A which lie towards the windows. The combs are fixed to metal shanks, which pass through a stout horizontal bar of ebonite. One of these shanks terminates in a couple of balls at E, and the other carries a sliding electrode D with a long ebonite handle. The framework which carries the horizontal ebonite bar and supports the fixed plates, &c., will be understood from the figure.

The machine, as originally constructed by Holtz, contained only the parts we have described. Poggendorff doubled all the parts (except, of course, the electrodes D and E). The figure represents Ruhmkorff ’ s modification of this construction. Behind the fixed disc there is another fixed disc, with windows and armatures like the first, and, beyond that, another movable disc mounted on the axis F. The combs are double, as will be seen from the figure. To start the machine, D and E are brought together, and one of the armatures (or one pair), say the right hand one, is electrified in any manner, let us say positively, and the disc set in rotation. After a little time a hissing noise is heard, and the machine becomes sensibly harder to turn, as if the disc were moving through a resisting medium. If the room be dark, long curved pencils of blue light will now be seen issuing from the points of the left hand comb, and running along the surface of the disc in a direction opposite to its motion, while little stars shine upon the points of the right-hand comb. After this state has been reached, the balls D, E may be separated, and a continuous series of brush discharges will take place between them, even when the distance is very considerable. If two Leyden jars, L, L, be hung upon the conductors which support the combs, the outer coatings being connected by a conductor M, then a succession of brilliant and sonorous sparks will take the place of the brushes. Instead of using the two jars L, L, we may connect D and E with the internal and external armatures of a condenser; it will then be found that, as we augment the capacity of the condenser (the angular velocity of the disc being constant), the frequency of the sparks diminishes, while their brilliancy increases. If we insert a high resistance galvanometer between D and E, it will indicate a current flowing from D to E, the intensity of which, under given atmospheric conditions and given state of the machine, will vary as the angular velocity, being independent, within very wide limits, of the resistance between D and E.

It is not difficult to give a general account of the action of this machine, although it is very hard to assign the precise importance of the individual parts, very slight modifications of which greatly affect the efficiency. Suppose D and E in contact; the righthand armature, charged +, acts by induction on the right-hand comb, causing - electricity to issue from the points upon the disc. At the same time the positive electricity of the right comb passes through DE to the left comb, and issues from its teeth upon the parts of the disc at the other end of the horizontal diameter. This + electricity electιifies the left armature-by induction, + electricity issuing from the blunt point upon the further side of the rotating disc. The charges thus deposited on the disc are carried along, so that the upper half is electrified - on both sides, and the lower half + on both sides, the sign of the electrification being reversed as the disc passes between the combs and the armature by the electricity issuing from the combs and from the armatures. If it were not for dissipation in various ways, the electrification everywhere would obviously go on increasing; but in practice a stationary condition is soon attained, in which the loss from the armatures is just balanced by the gain owing to the action of the blunt points. After this, both sides of the disc are similarly electrified, the upper half always -, the lower always +; + electricity continually issuing from the points of the right comb, - electricity from the points of the left. This is, of course, accompanied by a current of + electricity from right to left through DE.

The machine of Holtz, as we have described it, is somewhat uncertain in its action in our moist climate; but a slight modification of it gives excellent results. Upon the axis X is fixed a disc of ebonite, large enough just to reach between the paper armatures. This disc is fitted with a small rubber attached to the frame of the apparatus, and forms a small electric machine, which keeps the armatures continually electrified. The whole is inclosed in a glass case, with a beaker of sulphuric acid to dry the air. There is a machine of this kind at present in the Cavendish laboratory at Cambridge, which never fails when the auxiliary apparatus is at all in good order.

A very remarkable phenomenon often occurs when the electrodes of Holtz’s machine are in connection with the armatures of a condenser of considerable capacity, and are so far separated that a spark does not pass. The machine charges the condenser up to a certain point, and then the condenser discharges along the surface of the disc. If the experiment be conducted in a dark room, a flash of light will be seen to pass along the surface of the disc, and thereafter it will be observed that the long positive brushes have shifted from one comb to the other; after a little the condenser discharges again, and the brushes will now be seen in their old place, and so on. This phenomenon, though interesting to study, is often inconvenient in practice. To prevent it, Holtz introduced the diagonal conductor which is seen on many machines. For an account of this, and for other details concerning these machines, we refer the reader to Mascart, t. ii. § 847 sqg., whose account of the more obvious principles of this apparatus is among the most lucid we have seem His account of the experiment of causing one Holtz’s machine in action to turn the disc of another by the electrical reaction is of peculiar interest.

Electromagnetic Induction Machines.—The type of these is the induction coil or inductorium, sometimes called Ruhmkorff ’ s coil, after the great Parisian instrument-maker who first brought the instrument to perfection. The object of such machines is to obtain great electromotive force from sources which furnish large quantities of electricity, but have only small electromotive force.

The principles on which the action is founded has been sufficiently indicated above in our section on the induction of electric currents. We have also given in the Historical Sketch (p. 12) some notices of the literature of the subject; a brief enumeration of the essential parts of the machine is all that is necessary here.

We have first the primary coil—of thick wire and few windings, so as to have a small resistance and a small coefficient of self induction; the secondary coil surrounding the primary is of thin wire ( ∣⅛mm. or so), with many windings, the length in large machines being often 100,000 metres. In order to avoid the danger of disruptive discharge between parts of the insulated wire, the coil is divided up by insulating septa, so that parts at very different potentials are separated. In the centre of the primary is placed a bundle of iron wires; this greatly strengthens the action, and a good deal depends on the quality of the iron, which should be very soft. The interrupter is simply a lever, worked by the coil itself or by an electromagnet separate from the coil, by means of which the circuit of the primary is made and broken automatically. A variety of forms have been given to the part of the apparatus; the interrupter of Foucault is a veιy common one. For some purposes a break driven by clock-work is used. The condenser, a veιy important part of the apparatus, is made of a number of sheets of tinfoil, interleaved with sheets of oiled silk or varnished paper. One set of leaves of the condenser is connected with one side of the break, and the alternate set with the other side. The function of the condenser is to provide a way for the electricity when the circuit is broken, and thus to prevent the intense spark of the extra current in the primary, which destroys the contact surfaces of the break, and, what is worse, prolongs the fall of the primary current, and thereby reduces the average electromotive force of the induction current.

Other devices have been tried for effecting the same object as the condenser, such as inserting a fine metallic wire or an electrolyte as an alternative circuit to the break; and these answer the purpose to a considerable extent. An important improvement affecting this part of the apparatus has recently been introduced by breaking the primary circuit between the poles of a magnet, the effect of which is that the spark is suddenly drawn aside (blown out as it were). A considerable increase of striking distance between the poles of the secondary results from this arrangement.

absolute measurements.

We have already indicated the considerations which determine the fundamental units in the two systems that have come into practical use. We ought now to explain how practical standards can be constructed to represent these fundamental units, or at least known multiples of them. It is necessary to have such standards in order that we may be able to measure electrical quantities in absolute measure by simple and expeditious methods of comparison, it being obviously impossible in practice to make absolute measurements directly on all occasions.

Eleetrostatieal System.—By means of Thomson’s absolute electrometer we can determine any electromotive force in absolute measure. In this way Thomson found the electromotive force of Daniell’s battery to be - 00374 C.G.S. electrostatical units.

By using the absolute electrometer (see art. Electrometer), or another that had been compared with it, we could by the method given above, p. 46, find a resistance (which was large enough to suit the method) in eleetrostatieal measure.

Then, having standards of electromotive force and resistance, we could easily measure a current in electrostatic measure by applying Ohm’s law. The same thing might be done by constructing the standard of quantity, which is the charge on an isolated sphere of unit radius charged to unit potential. By comparing the throw of a galvanometer when unit quantity is discharged through it with the deflection produced by any current, we could determine the latter in absolute measure by observing the time of oscillation of the galvanometer and the logarithmic decrement of its oscillation (see Maxwell, vol. ii. § 749).

Among the absolute measurements in the present system of units, we must not omit to mention Sir Wm. Thomson’s determinations of the dielectric strength of different thicknesses of air. From these, and from the measurement of the electromotive force of Daniell’s cell just mentioned, he concluded that a Daniell’s battery of 5510 elements would be competent to produce a spark between two slightly curved metallic surfaces at ⅛ of a centimetre asunder in ordinary atmospheric air.

Electromagnetic System.—The great majority of the absolute determinations hitherto made have reference to this system. We make no attempt here to instruct the reader concerning the details of this subject; such an attempt would lead us into technical particulars intelligible only to a few scientific men. We are fortunate, however, in being able to refer the English reader to two books which contain in a collected form all, or nearly all, the requisite information, viz. Maxwell's Electricity ami Magnetism, and the collected Reports of the Committee of the British Association on Electrical Standards.

As a specimen of the theoretical considerations involved, the reader may take Maxwell’s method for determining the coefficient of self-induction of a coil (given above, p. 80). If we know the value of L (in centimetres) from calculation, then equation (33) might be used to find x in absolute measure. This would not be a practicable method, inasmuch as the calculation of L would be difficult if not impossible; we might, however, determine L by comparison with a coefficient of mutual induction which could be calculated.

The earliest absolute measurement of the resistance of a wire (by Kirchhoff in 1849) was of the kind just alluded to; that is to say, it involved the comparison of a resistance with a coefficient of mutual induction, the time measurement being that of the period of oscillation of a galvanometer.

Weber used two methods,—(1) the method of transient currents, in which he measured the throw of a galvanometer caused by the current from an earth inductor of known area when it was turned about a vertical axis, so that the number of the earth’s lines of force through it increased from zero to a maxi mum; and (2) the method of logarithmic decrements, in which he observed the time of oscillation and the logarithmic decrement of a magnet in a galvanometer of known constant. In the last of these two methods the horizontal component of the earth’s horizontal force comes in directly, and the magnetic moment of the galvanometer magnet must be determined, which is a matter of great difficulty.

The determination of the British Association committee was carried out by Messrs Maxwell, Balfour Stewart, and Fleemιng Jenkin, and the result of it was the construction of a standard called the ohm, which professes to represent a velocity of an earth quadrant per second (10 9 ~~).—The method they used is due to Sir Win. Thomson. It consists essentially in causing a coil of wire of known dimensions to rotate about a vertical axis, and observing the deflection of a magnet of very small moment suspended at its centre.

In a recent determination, F. Kohlrausch has combined the two methods of Weber, and thereby avoided some of the difficulties which arise in either method used by itself. His value for the Earth quadrant resistance of Siemens’s mercury unit is 0 - 9∕ 17 Second '

According to Dehms and Hermann Siemens, the resistance of the coil called the ohm is equal to 1 ’ 0493 mercury units. According to Kohlrausch, therefore, the actual British Association standard is

1 ∙0196 barth quadrant ∩ χ a p so j ute measure . or , j n other words, the

Second determination of the British Association Committee is out by nearly

2 per cent.

Lorenz has, still more recently, made a determination of the value of the mercury unit in absolute measure. He causes a copper disc to rotate inside a coil of known dimensions. The two ends of a circuit C are kept in contact with the axis and circumference respectively of this disc. At two points A and B of C, the resistance between which is R, are attached the two terminals of the coil of wire, in circuit with which is also a battery. A sensitive galvanometer is placed in the circuit C, and the angular velocity of the disc is adjusted till this galvanometer indicates no current. If n be the number of revolutions per second, and E the electromotive force of induction per unit of inducing current, calculated from the dimensions of the coil, then the resistance R is equal to nE in electromagnetic measure.

The result obtained by Lorenz for the value of the mercury unit is ‘9337 barth_quadrant. ∩ 1 ∣ s wou ∣q ma fc e the value of the B. A.

Second

,⅛r<i∙8797 ⅜⅛ quadrant.

Second

There is thus considerable discordance between the different results. It is a curious fact that the mean of the result of Kohlrausch and Lorenz gives for the value of the B. A. standard •9996 Earth quadrant , p res ] 1 determinations are, however, in Second progress, and it is to be hoped that the doubt which hangs over the matter will be dispelled.

Besides these methods, there is yet another of a totally different character, originally suggested by Thomson in 1851, in his paper on the “Mechanical Theory of Electrolysis.” This method consists in measuring the amount of heat developed in a wire by a current the square of whose strength is known in electromagnetic measure. If we know the mechanical equivalent of heat with sufficient accuracy, we can calculate from these results the resistance of the wire in absolute measure by means of Joule’s law. Measurements of this nature have been made by Von Quintus Icilius, Joule, and H. Weber.

We can, by means of a tangent galvanometer, find the value of any current in electromagnetic measure (see art. Galvanometer). If the resistance of the circuit be found, by comparison with the ohm or other absolute standard, we can determine the value of the electromotive force in the circuit by Ohm's law. Measurements of this kind have been made by Bosscha, by Von Waltenhofen, F. Kohlrausch, and Latimer Clark. The results of Kohlrausch for the cells of Daniell and Grove, when no current is passing, are 1138 ×10 5 and 1942 ×10 5 C.G.S. units respectively. Latimer Clark gives 1110 × 10“ and 1970 × 10“ for the same constants. The results, of course, depend on the constitution of the cells.

Taking the number of electromagnetic units in an electrostatic unit to be 3 × 10 ]0 , we get from Thomson’s electrostatic measurements for the electromotive force of Daniell’s element 1120 × 10 5 in C.G.S. units. The agreement among the different results is so far good.

The determination of the electrochemical equivalent of some elementary substance in this system of units is of great importance. Determinations exist by Weber, Bunsen, Casselmann, Joule, and F. Kohlrausch. The result of the last is no doubt the best, as he combined with his voltametric experiments a determination of the horizontal component of earth’s magnetic force, which is the most uncertain factor in the result. According to his result, one C.G.S. unit of electricity deposits - 011363 (± - 000002) gm. of silver. From this we get for the electrochemical equivalent of water •0009476.

Ratio of Electrostatic to Electromagnetic Unit.—Jf we measure the same quantity of electricity first in electrostatic and then in electromagnetic measure, the fundamental units of mass, length, and time being the same in both cases, the ratio of the two measures will vary directly as the magnitude of the unit of length, and inversely as the magnitude of the unit of time adopted. This ratio may therefore be regarded as a velocity which will remain the same whatever three fundamental units we adopt.

This velocity was found by Weber and Kohlrausch by the direct process of measuring the same quantity of electricity, first in terms of the one unit and then in terms of the other. This result was

31 X 10 9 —.

sec.

Five other methods will be found described by Maxwell, vol. ii. § 768 sqq. Two of these have actually been carried into execution,—one by himself, the other by Sir Wm. Thomson. The results for the fundamental velocity are 28'8 ×10 9 —h and sec.

28'2 × 10 9 ∏fi- - respectively.

Theories of Electrical Phenomena.

Throughout this article we have limited ourselves as much as possible to an exposition of the experimental facts of electricity. Where mathematical developments have occurred, they have in most cases been simply deductions from some principle or principles well established by experience. To have made our survey of the present state of electrical science complete, we ought to have added a section on the different attempts which have been made by the doctors of the science to penetrate a little farther into the secrets of the hidden mechanism by which electrical phenomena are brought about. But any attempt at a review of this kind must be relinquished. We refer the reader to our indications of the literature (Historical Sketch, p. 10). The most important work in this department lies at hand for the English reader in Professor Clerk Maxwell’s Treatise on Electricity anel Magnetism. Particularly important are his theory of electric displacement and its application to statical as well as to current electricity; his investigation of the stresses in the medium, by which the electrostatical forces on the one hand, and the electromagnetic forces on the other, may be produced; the application of the theory of displacement to the case of electrical equilibrium when the dielectric medium is not everywhere the same; the dynamical theory of the electromagnetic field; and the electromagnetic theory of light. Maxwell gives, at the end of his work, a most instructive summary of the different speculative theories. The student who desires to pursue this department farther will do well to master this summary at the outset. (G. CH.) Index.

The figures refer to the pages.

Absolute measurements, 103, 104; history of. Gauss, Weber, B.A. Committee, <tc., 15.

Accumulator, theory of, 34.

Alternating discbaιges with inductorium and Leyden jar, 65.

Ampère, electrodynamics, 10.

Amperes law, 70; experimental arrangements for showing electrodynamical action, 70.

Amperes theory, sketch of, 74; generalization of, 75.

Arc, voltaic, 58.

Batteries, 92-94; history of, 13; one-fluid and two-fluid —oxidizing agents in—local action—polarization, <fcc,,93.

Battery of Leyden jars, 35. Bound and free electricity, 35 Bowl, spherical, distribution on, Thomson, 33.

Brush, 63.

Capacity, coefficients of, 27. Cascade jars in, 35.

Cell of Daniell, different modifications of, 93, 94; of Grove, Dunsen, <tc., 94.

Chemical affinity, electrical measure of, Joule, 92.

Circuit, linear, Ohms law for, 42; action on, in magnetic field, 68.

Circular current, magnetic action of, 71.

Condensing electroscope, 34.

Conduction, Gray, 4; general equations of, 41.

Conductivity boxes, 45. Conductors, network of linear, Kirchhoff, 43; conjugate, 43.

Conjugate functions, 33. Conservation of energy, Joule,

Λc∙, 14.

Contact force, general law of, t∙3; Volta's experiments, 83; Volta’s law, 83; Kohl-r a usch's researches, 83; IIankeΓs experiment, 84; Thomson's demonstration, 85; Clifton's experiments, 85; uncertainty concerning, 85; from polarization, 86.

Contact of conductor with non-conductor, 100.

Contact of non-conductors, 99. Convection, electrolytic, Helmholtz, 87; of heat, electric, Thomson, 97; Tait's conjecture concerning, 97.

Convective discharge, 64, 66. Convecto-inductivβ machines,

Holtz, Töpler, Varley, Thomson, 101, 102, 103.

Coulomb, 9; his torsion balance, 18.

Current, electric, general phenomena and measure of, 40.

Currents, mutual action of, when parallel and when inclined, 70.

Decomposition of alkalis, Dary, 9; of water by electric current, Nicholson and Carlisle, 9; by electric spark, 9.

Dielectric strength of gases, <tc., Harris, Riess, <tc., 60; effect of pressure, <fcc., on, Harris, 61; Faraday s researches on, 61; Wiedeman n, Ruhlmann, 61; at high pressares, 62; minimum for vacuum, 62.

Differential galvanometer, 43. Discharge iu fluids, 65; in solids, 66; in gases, mag netic action on, De la Rire and Pliicker, 74.

Disruptive discharge, 59-66; theoretical considerations on, 60; progress of, 62.

Distribution, electrical, Coul-lomb, 19, 20, 22-24; general problem of, 27.

Doubler, Bennet's, Darwin, Cacallo, and Nicholson, 102.

Earth’s action on suspended current, 72.

Electricity, positive and negative, Dufay, 4; theory of, 17.

Electrics and non·electrics, Gilbert and Boyle, 3.

Electrodes, temperature of, in discharge through gases, 64.

Electrodynamics, theory of, Ampere, Weber, Neumann, Helmholtz, Maxicell, 10, 66-74.

Electrodynamometer, use of, in measuring electrolytic resistance, 49; Weber's experiments with, 71; Ampere's theory verified by means of, 71.

Electrokinetic energy, 81; Thomson s theory of, 76.

Electrolysis, Faraday, <fcc., 13, Electrolytes, Ohm's law for, 47; Faraday’s law of conduction for, 47; his law of electrochemical equivalents, 47; polarization and transition resistance with, 47; resistance of, Horsford, Beetz, Paalzow, Kohlrausch and Nippoldt, Ewing and Macgregor, 48-50.

Electromagnetic engines, history of, 10.

Electromagnetic rotation, 72; discovered by Faraday, 10; his apparatus, 72; Ampere’s theory of, 73; different apparatus for, 73; of fluids, 73; of electric discharge, 74.

Electromagnetism and electrodynamics, 66-75.

Electromotive force, origin of, 83-103; measurements of Poggendorff and Clark, 86; dynamical theory of, Thomson, 90; calculated from chemical data, 90; limit of, in electrolysis, 90; question as to seat of, 99.

Electromotive series — two metals and one liquid, two liquids, one metal and two liquids, <tc,, 86.

Electrophorus, Volta, 101.

Electrostatical theory, recent history of, Green, Thomson, Gauss, <tc., 15.

Electrostatics, mathematical theory, 24-36.

Electrostatics, experi ment al, recent history of, Faraday, Harris, Riess, <tc., 14.

Element of circuit, action on, in magnetic field, 68.

Ellipsoids, distribution on, 30

Energy, electrokinetic, of two circuits, 76.

Energy, laws of, in voltaic circuit, Joule, 89; Favre and Silbermann, 89; Thomson, 90.

Energy, transformations of, in electric circuit, 54, <fcc.

Equilibrium, electrostatic, condition of, 26.

Faraday, induction of electric currents, 11.

Figures of Lichtenberg, of Karsten, 66.

Fishes, electrical, 8.

Floating battery of De la Rice, 72.

Force, electric, laws of, Coulomb, 20, 21.

Force, electrostatic, for any displacement, 29,30.

Franklin's researches, 6; portrait experiment, 58.

Friction of powders, <fcc., Armstrong’s machine, 101.

Frictional electricity, contact theory of, 99.

Frictional electromotive series, Wilcke, Faraday, Riess, 100; Peclet's experiments, 100.

Frictional machines, Ramsden, Le Roy, Nairne, 100, 101.

Galvanometers, history of, 13; differential, 43.

Gas battery, Groce, 87.

Gases, rarefied, light effects in, 64. See also Dielectric strength.

Glow, 63.

Glowing of wires, 58. Gore's railway, 58.

Greene theorem, 29.

Heat, local, at electrodes, Joule, 90; Favre and Silbermann, Bosscha, 91; theories of Thomson, Joule, Bosscha, 92.

Heating effects, 55-59; general law of, 55; from discharge of statical electricity, 56; from constant current in metals and electrolytes, 56; reversible at junctions, <fcc., 57; general theory of, 57.

Images, electric, Thomson, 32; formed by surface electrification, 66.

Induction, Canton's discovery, 7, history of, 11; coefficients of, 27; through a surface, 25; between two fixed circuits, 81; in masses of metal, Pliicker, Foucault, 82.

Induction of electric currents, 77 -83; Faraday s laws and Maxwell's statements, 75; deduction of laws from conservation of energy by Helmhιltz and Thomson, 75; of two circuits, 76; coefficients of, 76; Neumann’s theory of, 76; Lenz's law of, 76; effect of material and thickness of wire, 76; effect of medium, 77; experiments with electrodynamometer, Weber, ∏; Felici's researches on, 77; unipolar, 77, 78; coils with iron core, 78; physiological effect of currents, 78; by statical discharge, 78; currents of higher orders, 79.

Induction, self, 76; Jenkin's observation, 79; Faraday's researches and theory, 79; Edlund's results 79; coefficient of, to measure, Maxwell, 80; calculationsand experiments of Helmholtz, 80.

Induction coil or inductorium, 103.

Inductive capacity, specific, Faraday, 36; Siemens, Gau-

gain, Gibson and Barclay, Boltzmann, 37; Schiller, Silow, Boltzmann, 38.

Inducto-convective machines, 101.

Inversion, electric, Thomson, 33.

Iron, action of soft, 75.

Joule's law for heating effect, 56.

Kite experiment, Franklin, 6.

Lenz, law of induction, 11. Level surfaces, theory of, 25. Leyden jar, Muschenbroeck, <fcc., 5; theory of, 35.

Lichtenberg's figures, 66.

Light, electric, Guericke, 4; in Torricellian vacuum. Hauksbee, 4; phenomena, 59.

Lines of force, theory of, 26.

Local action in batteries, 93.

Magnetic pole, action of current on, 70.

Magnetism of rotation, Arago's discovery, 11; his experiment, 82; Faraday's explanation, 82; mathematical investigations, 83.

Magnetization by current, Arago and Dary, 10.

Magneto-electric machines, Pixti, <fcc., 12.

Mance's method of measuring battery resistance, 50.

Measurements, absolute, 103- 104.

Mechanical effects, 65. Medium, insulating, 36-40. Melting of wires, 58.

Multiple arc, 43,

ΛVumαnn, F. E., theory of induction, 11.

Oersted, magnetic action of current, 10.

Ohm's law, 40-43; history of, 12; for electrolytes, 47.

oscilUtions,electrical,7%omson, 81; experiments of Feddersen, Schiller, <fcc., 82.

Peltier effect, 57.

Phenomena, fundamental, 16.

Physiological effects of electric currents, 78.

Points and edges, density at, 31.

Poisson, 9.

Polarization, history of, 14; varieties of, 86; in batteries, 93; by gases, 87; maximum of, on what it depends, 88; decay of, 88; numerical results concerning, 88; un-polarizable electrodes, 89.

Potential, electrostatic, theory of, 24-27; coefficients of, 28; fall of, in voltaic circuit, 42. ·

Potential of magnetic shell, 67; of two circuits and of circuit on itself, Neumann, 76.

Potential energy, theorem of mutual, 28.

Potential energy of system, 29. Pyroelectricity, Canton, Wilcke, Bergman, &c., 96; early history of, 8.

Quantity, electric, 18, 19.

Residual discharge, Kohlrausch, 39; Maxwell, Hopkinson, 40.

Resistance, measurement of, 43-46; history of measurements, 13; of battery, 50; specific, general tabie of, 53; of transition, 87.

Resistance in general, on, 51; of metals and alloys, data concerning, 51; of electrolytes, data concerning, 52.

Resistance boxes, 45.

Resistances, measurement of small, 45, 46; ef great, with electrometer, 46.

Rheostat and rheochord 45.

Screens, electrical, 29.

Shell, magnetic, representing the action of a current, 67.

Sine inductor, experiments with, 48, 49

Solenoid, 7L Spark, 62.

Sphere with given force, 31. Spheres, two influencing, Poisson, 3L

Standards of resistance, 44. Surface electrification, 66. Synthetical method, Green, 32. Systems, internal and external, 29.

Tension defined, 60; limiting 60; positive and negative, 61, 62.

Theories, speculative, of electrical phenomena, 104; contact and chemical, of voltaic circuit, 14.

Theory, one-fluid, Franklin, 6; one-fluid, Cavendish, S; one-fluid, xEpinus, 8; two·fluid, Symmer, 7,

Thermoelectric diagram, Tait, 97.

Thermoelectric inversion, Cumming, 96.

Thermoelectric sériée, Seebeck, 94.

Thermoelectricity, 94-99; history of, Secbeck, Cumming, Peltier, Thomson, <fcc., 11; Seebeck's discovery, 94.

Thermoelectrometer, Harris and Riess, 55.

'lhermoelectromotive force, order of magnitude of, 95; results of Magnus, 95; effects of strain, <tc., on, Thomson and Le Roux, 95; in circuits of one metal, 96; Thomson's theory of, 96,97; Tait's addition to Thomson's theory, 97; his experiments and results, 98.

Thomson effect, 57, 97. Torsion balance of Coulomb, 18.

Transmission of electricity, Watson, 5.

Trevelyan rocker, 58.

Unit, electromagnetic, of current strength, 71; of electromotive force, 75.

Units, electrostatic and electromagnetic, 41.

vector potential of magnetic sheU, 68.

velocity of electricity, Wheatstone, 63.

Volatilization of wires, 4tc., 58.

water-dropping electric machine, Thomson, 102.

Weber, 10; his experiments with electrodynamometer, 71.

Wheatstone's bridge, 44.