Earth

EARTH, Figure of the. The determination of the figure of the earth is a problem of the highest importance in astronomy, inasmuch as the diameter of the earth is the unit to which all celestial distances must be referred. Reasoning, doubtless, from the uniform level appearance of the horizon in any situation in which a spectator can be placed—the variations in altitude of the circumpolar stars as one travels towards the north or south, the disappearance of a ship standing out to sea, and perhaps other phenomena— the earliest astronomers universally regarded this earth as a sphere, and they endeavoured to ascertain its dimensions. Aristotle relates that the mathematicians had found the circumference to be 400,000 stadia. But Eratosthenes appears to have been the first who entertained an accurate idea of the principles on which the determination of the figure of the earth really depends, and attempted to reduce them to practice. His results were very inaccurate, but his method is the same as that which is followed at the present day—depending, in fact, on the comparison of a line measured on the earth’s surface with the corresponding arc of the heavens. He observed that at Syene in Upper Egypt, on the day of the summer solstice, the sun was exactly vertical, whilst at Alexandria at the same season of the year its zenith distance was 7° 12', or one-fiftieth of the circumference of a circle. He assumed that these places were on the same meridian; and, reckoning their distance apart as 5000 stadia, he inferred that the circumference of the earth was 250,000 stadia. A similar attempt was made by Posidonius, who adopted a method which differed from that of Eratosthenes only in using a star instead of the sun. He obtained 240,000 stadia for the circumference. But it is impossible to form any correct opinions as to the degree of accuracy attained in these measures, as the length of the stadium is unknown. Ptolemy in his Geography assigns the length of the degree as 500 stadia.

The Arabs, who were not inattentive to astronomy, did not overlook the question of the earth’s magnitude. The caliph Almamoum, 814 A.D., having fixed on a spot in the plains of Mesopotamia, despatched one company of astronomers northwards and another southwards, measuring the journey by rods, until each found the altitude of the pole to have changed one degree. But the result of this measurement does not appear to have been very satisfactory. From this time the subject seems to have attracted no attention until about 1500, when Fern el, a Frenchman, measured a distance in the direction of the meridian near Paris by counting the number of revolutions of the wheel of his carriage as he travelled. His astronomical observations were made with a triangle used as a quadrant, and his resulting length of a degree was by a happy chance very near the truth.

The next geodesist, Willebrυrd Snell, took an immense step in the right direction by substituting a chain of triangles for actual linear measurement. The account of this operation was published at Leyden in 1617. He measured his base line on the frozen surface of the meadows near Leyden, and measured the angles of his triangles, which lay between Alkmaar and Bergen-op-Zoom, with a quadrant and semicircles. He took the precaution of comparing his standard with that of the French, so that his result was expressed in toises (the length of the toise is about 6 - 39 English feet). The work was recomputed and reobserved by Muschenbroek in 1729.

In 1637 an Englishman, Richard Norwood, published his own determination of the figure of the earth in a volume entitled The Seaman's Practice, cοntayniηg a Fundamentali Probleme in Navigation experimentally verified, namely, touching the Compasse of the Earth and Sea and the quantity of a Degree in our English Measures. It appears that he observed on the 11th June 1633 the sun’s meridian altitude in London as 62° Γ, and on June 6, 1635, his meridian altitude in York as 59° 33 ’. He measured the distance between these places along the public road partly with a chain and partly by pacing. By this means, through compensation of errors, he arrived at 367,176 feet for the degree—a very fair result.

The application of the telescope to circular instruments was the next important step in the science of measurement. Picard was the first who in 1669, with the telescope, using such precautions as the nature of the operation requires, measured an arc of meridian. He measured with wooden rods a base line of 5663 toises, and a second or base of verification of 3902 toises; his triangulation extended from Mal voisine, near Paris, to Sourdon, near Amiens. The angles of the triangles were measured with a quadrant furnished with a telescope having cross-wires in its focus. The difference of latitude of the terminal stations was determined by observations made "with a sector on a star in Cassiopeia, giving 1° 22' 55'' for the amplitude. The terrestrial measurement gave 78,850 toises, whence he inferred for the length of the degree 57,060 toises.

Hitherto geodetic observations had been confined to the determination of the magnitude of the earth considered as a sphere, but a discovery made by Richer turned tho attention of mathematicians to its deviation from a spherical form. This astronomer, having been sent by the Academy of Sciences of Paris to the island of Cayenne, in South America, for the purpose of determining the amount of terrestrial refraction and other astronomical objects, observed that his clock, which had been regulated at Paris to beat seconds, lost about two minutes and a half daily at Cayenue, and that in order to bring it to measure mean solar time it was necessary to shorten the pendulum by more than a line. This fact, which appeared exceedingly curious, and was scarcely credited till it had been confirmed by the subsequent observations of Varin and Deshayes on the coasts of Africa and America, was first explained in the third book of Newton’s Principia, who showed that it could only be referred to a diminution of gravity arising either from a protuberance of the equatorial parts of the earth and consequent increase of the distance from the centre or from the counteracting effect of the centrifugal force. About the same time, 1673, appeared the work of Huyghens entitled De Horologio Oscillator™, in which for the first time were found correct notions on the subject of centrifugal force. It does not, however, appear that they were applied to the theoretical investigation of the figure of the earth before the publication of Newton's Principia. In 1690 Huyghens, following up the subject, published his treatise entitled De Causa Gravitatis, which contains an investigation of the figure of the earth on the supposition that the attraction of every particle is towards the centre.

Between 1684 and 1718 J. and D. Cassini, starting from Picard’s base, carried a triangulation northwards from Paris to Dunkirk and southwards from Paris to Collioure. They measured a base of 7246 toises near Perpignan, and a somewhat shorter base near Dunkirk; and from the northern portion of the arc, which had an amplitude of 2’ 12' 9", obtained for the length of a degree 56,960 toises; while from the southern portion, of which the amplitude was 6° 18' 57'', they obtained 57,097 toises. The immediate inference from this was that, the degree diminishing with increasing latitude, the earth must be a prolate spheroid. This conclusion was totally opposed to the theoretical investigations of Newton and Huyghens, and created a great sensation among the scientific men of the day. The question was far too important to be allowed to remain unsettled, and accordingly the Academy of Sciences of Paris determined to apply a decisive test by the measurement of arcs at a great distance from each other. For this purpose some of the most distinguished members of their body undertook the measurement of two meridian arcs—one in the neighbourhood of the equator, the other in a high latitude; and so arose the celebrated expeditions of the French Academicians. In May 1735, MM. Godin, Bouguer, and De la Condamine, under the auspices of Louis XV., proceeded to Peru, where, assisted by two Spanish officers, after ten years of laborious exertion they measured an arc of 3° 7' intersected by the equator. The second party consisted of Maupertuis, Clairaut, Camus, Lemonnier, and Outhier, who reached the Gulf of Bothnia in July 1736; they were in some respects more fortunate than the first party, inasmuch as they completed the measurement of an arc near the polar circle of 57'amplitude and returned to Europe within sixteen months from the date of their departure.

The measurement of Bouguer and De la Condamine was executed with great care, and on account of the locality, as well as the manner in which all the details were conducted, it has always been regarded as a most valuable determination. The southern limit was at a place called Tarqui, the northern at Cotchesqui. A base of 6272 toises was measured in the vicinity of Quito, near the northern extremity of the arc, and a second base of 5260 toises near the southern extremity. The mountainous nature of the country made the work very laborious, in some instances the difference of heights of two neighbouring stations exceeding a mile. The difficulties with which the observers had to contend were increased by the opposition of the more ignorant of the inhabitants, and they were at times in danger of losing their lives. They had also much trouble with their instruments, those with which they were to determine the latitudes proving untrustworthy. But their energy and ingenuity were equal to the occasion, and they succeeded by simultaneous observations of the same star at the two extremities of the arc in obtaining very fair results. The whole length of the arc amounted to 176,945 toises, while the difference of latitudes was 3° 7' 3”. In consequence of a misunderstanding that arose between De la Condamine and Bouguer, their operations were conducted separately, and each wrote a full and interesting account of the operation. Bouguer’s book was published in 1749; that of De la Condamine in 1751. The toise used in this measure was ever after regarded as the standard toise, and is always referred to as the Toise of Peru.

The party of Maupertuis, though their work was quickly despatched, had also to contend with great difficulties. They were disappointed in not being able to make use of the small islands in the Gulf of Bothnia for the trigonometrical stations, and were forced to penetrate into the forests of Lapland. They commenced operations at Tornea, a city situated on the mainland near the extremity of the gulf. From this, the southern extremity of their arc, they carried a chain of triangles northward to the mountain Kittis, which they selected as the northern terminus. In the prosecution of this work they suffered greatly from cold and the bites of flies and gnats. The latitudes were determined by observations with a sector (made by Graham) of the zenith distance of a and δ Draconis. The base line was measured on the frozen surface of the river Tornea about the middle of the arc; two parties measured it separately, and they differed by about 4 inches. The result of the whole was that the difference of latitudes of the terminal stations was 57' 29" ’ 6, and the length of the arc 55,023 toises. In this expedition, as well as in that to Peru, observations were made with a pendulum to determine the force of gravity; and these observations coincided with the geodetical results in proving that the earth was an oblate and not prolate spheroid.

In 1740 was published in the Paris Mémoires an account, by Cassini de Thury, of a remeasurement by himself and Lacaille of the meridian of Paris. With a view to determine more accurately the variation of the degree along the meridian, they divided the distance from Dunkirk to Collioure into four partial arcs of about two degrees each, by observing the latitude at five stations. The anomalous results previously obtained by J. and D. Cassini were not confirmed, but on the contrary the length of the degree derived from these partial arcs showed on the whole an increase with increasing latitude. In continuation of their labours, Cassini and Lacaille further measured an arc of parallel across the mouth of the Rhone. The difference of time of the extremities was determined by the observers at either end noting the instant of a signal given by flashing gunpowder at a point near the middle of the arc.

While at the Cape of Good Hope in 1752, engaged in various astronomical observations, Lacaille measured an arc of meridian of 1° 13' 17'', which gave him for the length of the degree 57,037 toises—an unexpected result, which has led to the modern remeasurement of the arc by Sir Thomas Maclear.

Passing over the measurements made between Rome and Rimini and on the plains of Piedmont by the Jesuits Boscovich and Beccaria, and also the arc measured with deal rods in North America by Messrs Mason and Dixon, we come to the commencement of the English triangulation. In 1783, in consequence of a representation from Cassini de Thury on the advantages that would accrue to science from the geodetic connection of Paris and Greenwich, General Roy was with the king’s approval appointed by the Royal Society to conduct the operations on the part of this country,—Count Cassini, Mechain, and Delambre being appointed on the French side. And now a precision previously unknown was brought into geodesy by the use of Ramsden’s splendid theodolite, which, was the first to make the spherical excess of triangles measurable. The wooden rods with which the first base was measured were speedily replaced by glass rods, which again were rejected for the steel chain of Ramsden. The details of this operation are fully given in the Account of the Trigonometrical Survey of England and Wales. Shortly after this, the National Convention of France, having agreed to remodel their system of weights and measures, chose, as applicable to all countries, for their unit of length the ten-millionth part of the meridian quadrant. In order to obtain this length precisely, the remeasurement of the French meridian was resolved on, and deputed to Delambre and Mechain. The details of this great operation will be found in the Base du Système Métrique Décimale. The arc was subsequently extended by MM. Biot and Arago to the island of Iviza.

The appearance in 1838 of Bessel’s classical work entitled Gradmessung in Ostpreussen marks an era in the science of geodesy. Here we find the method of least squares, a branch of the theory of probabilities, applied to the calculation of a network of triangles and the reduction of the observations generally. This work has been looked on as a model ever since, and probably it will not soon be superseded as such. The systematic manner in which all the observations were taken with the view of securing final results of extreme accuracy is admirable. The triangulation, which is a small one, extends about a degree and a half along the shores of the Baltic in a N.N.E. direction. The compound bars with which he measured his base line may be understood by the following brief description. On the surface of an iron bar two toises in length is laid a zinc bar, both being very perfectly planed and in free contact— the zinc bar being slightly shorter than the iron bar. They are united at one end only, and as the temperature varies the difference of length of the bars as seen at the other end varies; this difference of length is a thermometrical indication whereby a correction for temperature can be applied to the bars so as to reduce their length to that at the standard temperature. The bars in measuring were not allowed to come into contact, but the intervals left were measured by the interposition of a glass wedge. The results of all the comparisons of the four measuring rods with one another, and with the standards, are elaborately worked out by least squares. The angles were observed with theodolites of 12 and 15 inches diameter, and the latitudes determined by means of the transit instrument in the prime vertical—a method much used in Germany. The formulae employed in the reduction of the astronomical observations are very elegant. The reduction of the triangulation waβ carried out in the most thorough manner,—the sum of the squares of all the actual theodolite observations being made a minimum. As it is usual now to follow this method (sometimes only approximately) in all triangulations where great precision is required, we here give a brief description of the method. The equations of condition of a triangulation are those which exist between the supernumerary observed quantities and their calculated values, for, after there are just sufficient observations to fix all the points, then any angle that may be subsequently observed can be compared with its calculated value. If a triangulation consist of n + 2 points, two of which are the ends of a base line, then to fix the n points 2n angles suffice; so that if m be the actual number of angles really observed, the triangulation must afford m - 2n equations of condition. To show how these arise, suppose that from a number m of fixed points A, B, C . . . a new point P is observed, which m points are again observed from P, then there will be formed m - 1 triangles, in each of which the sum of the observed angles is = 180° + the spherical excess; this gives at once m - 1 equations of condition. The m — 2 distances will each afford an equation of the form

PC PB PA

PB ’ PA PC ’

not, however, limited to three factors. Should P observe the m points and not be observed back, there will be m - 3 equations of the above form (they are called side equations). In a similar manner other cases can be treated. In practice the ratios of sides are replaced by the ratios of the sines of the corresponding opposite angles. To each observed angle a symbolical correction is applied, so that if α be an observed angle and a + x the true or most probable angle, sin (α + z) = sin α(1 + x cot a), x being a small angle whose square is neglected. Thus the side equation takes the form β + β λ x χ + β i x 2 + . . . β 1 x t = 0. In the case of equations formed by adding together the three observed angles of a triangle the co-efficients are of course unity. The problem then is this: Given n equations

β + 3 r r 1 + β. 2 x 2 + . . . β m r m = 0 β' + j 3 1 'z 1 + β 3 'x 2 + . . . βfx m = 0 β"+ βfx 1 + β i "x a + . . . β m "r m = 0

between m(m>n) unknown quantities x 1 . . .x m , which are the corrections (expressed in seconds of arc) to the observed angles, it is required to determine these quantities so as to render the function w 1 x 1 2 + tv 2 x 2 2 + w. 0 x 3 2 + . . . w m x m 2 a minimum, where w 1 . . . w m are the weights of the determinations of the angles to which the corresponding corrections belong. The corrections x l . . . x η fulfilling this condition of minimum have, according to the theory of least squares, a higher probability than any other system of corrections that merely satisfy the equations of condition. Multiply the n equations by multipliers λ 1 , λ 2 , . . . λ n , and we obtain by the theory of maxima and minima m equations w 1 x 1 ≈ β l λ 1 + β 1 ’ λ i + βfλ i + . . .

= 3aM ^t h 2 + β∙> λj + . l j wmrm — 3∙tλj + βmλ 2 + ß M λ-3 + . .

The values of x i . . . x m obtained from these equations are to be substituted in the original equations of condition, and then there will be n equations between the n multipliers λ 1 . . . λ n . These being solved, the numerical values of λ 1 . . . λ n will be obtained, and on substituting these in the last equations written down, the values of aq . . . ir ιn will follow. The process is a long and tedious one; but it is inevitable if we wish very good results.

The great meridian arc in India was commenced by Colonel Lambton at Punnoe in latitude 8° 9'. Following generally the methods of the English survey, he carried his triangulation as far north as 20° 30'. The work then passed into the able hands of Sir George (then Captain) Everest, who continued it to the latitude of 29° 30'. Two admirably written volumes by Sir George Everest, published in 1830 and in 1847, give all the details of the vast undertaking. The great trigonometrical survey of India is now being prosecuted with great scientific skill by Colonel Walker, R.E., and it may be expected that we shall soon have some valuable contributions to the great problem of geodesy. The working out of the Indian chains of triangle by the method of least squares presents peculiar difficulties, but enormous in extent as the work is, it is being thoroughly carried out. The ten base lines on which the survey depends were measured with Colby’s compensation bars.

These compensation bars were also used by Sir Thomas Maclear in the measurement of the base line in his extension of Lacaille’s arc at the Cape. The account of this operation will be found in a volume entitled Verification and Extension of Lacaille’s Arc of Meridian at the Cape of Good Hope, by Sir Thomas Maclear, published in 1866. Lacaille ’ 3 amplitude is verified, but not his terrestrial measurement.

The number of stations in the principal triangulation of Great Britain and Irelaud is about 250. At 32 of these the latitudes were determined with Ramsden’s and Airy’s zenith sectors. The theodolites used for this work were, in addition to the two great theodolites of Ramsden which were used by General Roy and Captain Kater (and which are now in as good condition as when they came from the hands of the maker), a smaller theodolite of 18 inches diameter by the same mechanician, and another of 24 inches diameter by Messrs Troughton and Simms. Observations for determination of absolute azimuth were made with these instruments at a large number of stations; the stars α, δ, and λ Ursae Minoris and 51 Cephei being those observed, always at the greatest azimuths. At six of these stations the probable error of the result is under 0" - 4, at twelve under 0" ’ 5, at thirty-four under 0" - 7: so that the absolute azimuth of the whole network is determined with extreme accuracy. Of the seven base lines w hich have been measured, five were by means of steel chains and two with Colby’s compensation bars. This is a system of six compound bars self-correcting for temperature. The compound bar may be thus described. Two bars, one of brass and the other of iron, are laid side by side, parallel, and firmly united at their centres, from which they are free to expand or contract; at the standard temperature they are of the same length. Let AB be one bar, A ’ B' the other; draw a line through the corresponding extremities A, A' to P, aud a line through the other extremities B, B' to Q, make A'P = B'Q, AA' being = BB'. Now if A'P is to AP as the rate of expansion of the bar A'B' to the rate of expansion of the bar AB, then clearly the distance PQ will be invariable, or very nearly so. In the actual instrument P and Q are finely engraved dots at the distance of 10 feet apart. In the measurement the bars when aligned do not come into contact; an interval of six inches is left between each bar and its neighbour. This small space is measured by an ingenious micrometrical arrangement constructed on exactly the same principle as the bars themselves. The triangulation was computed by least squares. The total number of equations of condition for the triangulation is 920; if therefore the whole had been reduced in one mass, as it should have been, the solution of an equation of 920 unknown quantities would have occurred as a part of the work. To avoid this an approximation was resorted to; the triangulation was divided into twenty-one parts or figures; four of these, not adjacent, were first adjusted by the method explained, and the corrections thus determined in these figures carried into the equations of condition of the adjacent figures. The average number of equations in a figure is 44; the largest equation is one of 77 unknown quantities.

Airy’s Zenith Sector is too well known to need description. The vertical limb is read by four microscopes; altogether, in the complete observation of a star there are 10 micrometer readings and 12 level readings. In some recent observations in Scotland for latitude the Zenith Telescope has been used with very great success; it is very portable; and a complete determination of latitude, affected with the mean of the declination errors of two stars, is effected by two micrometer readings and four level readings. The observation consists in measuring with the telescope micrometer the difference of zenith distances of two stars which cross the meridian, one to the north and the other to the south of the observer at zenith distances which differ by not much more than 10' or 15', the interval of the times of transit being not less than one nor more than twenty minutes. The advantages are that, with simplicity in the construction of the instrument and facility in the manipulation, refraction is eliminated (or nearly so, as the stars are generally selected within 25° of the zenith), and there is no large divided circle. The telescope, which is counterpoised on one side of the vertical axis, has a small circle for finding, and there is also a small horizontal circle. This instrument is universally used in American geodesy.

The United States Coast Survey has a principal triangulation extending for about 9° 30' along the coast, but the final results are not yet published.

In 1860 was published F. G. Struve’s Arc du Méridien de 25° 20' entre le Danube et la Mer Glaciale mesuré depuis 1816 jusqu'en 1855. This work is the record of a vast amount of scientific labour and is the greatest contribution yet made to the question of the figure of the earth. The latitudes of the thirteen astronomical stations of this arc were determined partly with vertical circles and partly by means of the transit instrument in the prime vertical. The triangulation, a great part of which, however, is a simple chain of triangles, is reduced by the method of least squares, and the probable errors of the resulting distances of parallels is given; the probable error of the whole arc in length is ± 6 - 2 toises. Ten base lines were measured. The sum of the lengths of the ten measured bases is 29,863 toises, so that the average length of a base line is 19,100 feet. The azimuths were observed at fourteen stations. In high latitudes the determination of the meridian is a matter of great difficulty; nevertheless the azimuths at all the northern stations were successfully determined,—the probable error of the result at Fuglenoes being ÷ 0" ’ 53.

Mechanical Theory.

Newton appears to have been the first to apply his own newly-discovered doctrine of gravitation, combined with the so-called centrifugal force, to the question of the figure of the earth. Assuming that an oblate ellipsoid of rotation is a form of equilibrium for a homogeneous fluid rotating with uniform angular velocity, he obtained the ratio of the axes 229: 230, and the law of variation of gravity on the surface. A few years later Huyghens published an investigation of the figure of the earth, supposing the attraction of every particle to be towards the centre of the earth, obtaining as a result that the proportion of the axes should be 578:579. In 1740 Maclaurin wrote his celebrated essay on the tides, one of the most elegant geometrical investigations ever made. He demonstrated that the oblate ellipsoid of revolution is a figure which satisfies the conditions of equilibrium in the case of a revolving homogeneous fluid mass whose particles attract one another according to the law of the inverse square of the distance; he gave the equation connecting the ellipticity with the proportion of the centrifugal force at the equator to gravity, and he determined the attraction on a particle situated anywhere on the surface of such a body. Some few years afterwards Clairaut published (1743) his Théorie de la Figure de la Terre, which contains, among other results, demonstrated with singular elegance, a very remarkable theorem which establishes a relation between the ellipticity of the earth and the variations of gravity at different points of its surface. Assuming that the earth is composed of concentric ellipsoidal strata having a common axis of rotation, each stratum homogeneous in itself, but the ellipticities and densities of the successive strata varying according to any law, and that the superficial stratum has the same form as if it were fluid, he proves the very important theorem contained in the equation f-g . 5

-—- i, + e = δ ∙m, g 2

Where g, f are the amouhts of gravity at the equator and at the pole respectively, e the ellipticity of the meridian, and m the ratio of the centrifugal force at the equator to g. Clairaut also proved that the increase of gravity in proceeding from the equator to the poles is as the square of the sine of the latitude. This, taken with the former theorem, gives the means of determining the earth ’ 3 ellipticity from observation of the comparative force of gravity at any two places. Clairaut would seem almost to have exhausted the subject, for although much has been written since by mathematicians of the greatest eminence, yet, practically, very little of importance has been added. Laplace, himself a prince of mathematicians, who had devoted much of his own time to the same subject, remarks on Clairaut’s work that “the importance of all his results and the elegance with which they are presented place this work amongst the most beautiful of mathematical productions” (Todhunter’s History of the Mathematical Theories of Attraction and the Figure of the Earth, vol. i. p. 229).

The problem of the figure of the earth treated as a question of mechanics or hydrostatics is one of great difficulty, and it would be quite impracticable but for the circumstance that the surface differs but little from a sphere. In order to express the forces at any point of the body arising from the attraction of its particles, the form of the surface is required, but this form is the very one which it is the object of the investigation to discover; hence the complexity of the subject, and even with all the present resources of mathematicians only a partial and imperfect solution can be obtained, and that not without some labour. We may, however, here briefly indicate the line of reasoning by which some of the most important of the results we have alluded to above may be obtained. The principles of hydrostatics show us that if X, Y, Z be the components parallel to three rectangular axes of the forces acting on a particle of a fluid mass at the point x, y, z, then, p being the pressure there, and p the density, dp = p(Xdx + Υdy+ Zdz); and for equilibrium the necessary conditions are, that p (Xdx + Υdy + Zdx) be a complete differential, and at the free surface Xdx + Υdy + Zcfz = 0. This equation implies that the resultant of the forces is normal to the surface at every point, and in a homogeneous fluid it is obviously the differential equation of all surfaces of equal pressure. If the fluid be heterogeneous then it is to be remarked that for forces of attraction according to the ordinary law of gravitation, if X, Y, Z be the components of the attraction of a mass whose potential is V, then v , vj - 7j <ilV j dV, , <ZV, ∖dx + Υdy÷7idz — t , which is a complete differential. And in the case of a fluid rotating with uniform velocity, in which the so-called centrifugal force enters as a force acting on each particle proportional to its distance from the axis of rotation, the corresponding part of Xdx + Sidy + Z<fe is obviously a complete differential. Therefore for the forces with which we are now concerned Xcfø +Y<⅞∕ + ZcZz = <7U, where U is some function of x, y, z, and it is necessary for equilibrium that dp = pdU be a complete differential; that is, p must be a function of U or a function of p, and so also p a function of U. So that tZU = 0 is the differential equation of surfaces of equal pressure and density.

We may now show that a homogeneous fluid mass in the form of an oblate ellipsoid of revolution having a uniform velocity of rotation can be in equilibrium. It may be proved that the attraction of the ellipsoid z 2 +y 2 + z 2 (1 + ∈ 2 ) = c 2 (1 + ∈ 2 ) upon a particle P of its mass at x, y, z has for components

X = Ax, Y = Ay, Z = Cz, where a = 2π ∕ , (⅛ i ^ tan ~ le “ «0 C=4τp(^-l±≤ t ι m" 1 e)

Besides the attraction of the mas3 of the ellipsoid, the eentrifugal force at P has for components - xω 2 , - yω 2 , 0; then the condition of fluid equilibrium is

(A - ω 2 )xcfo + (A - ω i ')ydy + Czdz = 0, which by integrating gives

(A - ω 2 )(aj 2 + y 2 ) + C≈ 2 = constant.

This is the equation of an ellipsoid of rotation, and therefore the equilibrium is possible. The equation coincides with that of the surface of the fluid mass if we make a s _ C

A - w- - —„ ,

1 + e- ’ which gives

ω 2 3 + e 2 , 3

ö— ——3~ tan e - .

2πρ e i e 2

If we would determine the maximum value of ω from tiιis equation, we find that it corresponds to the value of e determined by the condition + -1 _ 9e + 7e 3

* “ (l+e 2 X9 + e 2 ) ’

hence it may be shown that if the angular velocity exceed ω 2

that calculated from = 0'2247, equilibrium is impossible for the form of an ellipsoid of revolution. If ω fall short of this limit, there are two ellipsoids which satisfy the condition of equilibrium; in one of these the eccentricity is greater and in the other less than 0 ’ 93. In the case of the earth, which is nearly spherical, we get by expanding the expression for ω 2 in powers of e 2 , rejecting the higher powers, and remarking that the ellipticity e = ∣e 2 , jΔ-± e2 =A β 2∙∏ 7 > 15 15 ‘

Now, if m be the ratio of the centrifugal force at the equator to gravity there, cω 2 . ω 2 _ 2 m m ⅜τrpc - Cω 2 ’ 2ιrp 3 1 + m

In the case of the earth it is a matter of observation that m = - 2 -g- 5 -, hence the ellipticity

5 1

i- 4 w ~ 2312 so that the ratio of the axes on the supposition of a homogeneous fluid earth is 230: 231, as announced by Newton.

Now, to come to the case of a heterogeneous fluid, we shall assume that its surfaces of equal density are spheroids, concentric and having a common axis of rotation, and that the ellipticity of these surfaces varies from the centre to the outer surface, the den⅛ty also varying. In other words, the body is composed m homogeneous spheroidal shells of variable density and ellipticity. On this supposition we shall express the attraction of the mass upon a particle in its interior, and then, taking into account the centrifugal force, form the equation expressing the condition of fluid equilibrium. The attraction of the homogeneous spheroid rc 2 + y 2 + 2 2 (1 + 2e) = c 2 (1 + 2e), where e is the ellipticity, of which the square is neglected, on an internal particle, whose co-ordinates are x —f, y — 0, z = h, has for its x and z components x -l' pz (1 -∣ e )' z '=√' ,i (1+ ι e )' th¾Y component being of course zero. Hence we infer that the attraction of a shell whose inner surface has an ellipticity e, and its outer surface an ellipticity e + de, the density being p, is expressed by

<ZX' = -5- · 7 τrpfde, dZ'=θ · J τrphde.

3 5 3 5

To apply this to our heterogeneous spheroid; if we put c l for the semiaxis of that surface of equal density on which is situated the attracted point P, and c 0 for the semiaxis of the outer surface, the attraction of that portion of the body which is exterior to P, namely, of all the shells which inclose P, has for components both e and p being functions of c. Again the attraction of a homogeneous spheroid of density p on an external point f, h has the components

X"= j c3 < 1 + 2e ) - λec5 J

z "= ~-P c ∖ l + 2e) - λ'ec 5 1 x 3 4h 2 -f 2 . l 3 2h 2 - 3∕ 2 , o λo „

where λ = 5 · —74— ’ λ = 5 * —74~, αnd ** 2 =∕ 2 + ^ 2 . Now e being considered a function of c, we can at once express the attraction of a shell (density p) contained between the surface defined by c + dc, e + de and that defined by c, e upon an external point; the differentials with respect to c, viz. <7X" <7Z", must then be integrated with p under the integral sign as being a function of c. The integration will extend from c≈0 to c = c 1 . Thus the components of the attraction of the heterogeneous spheroid upon a particle within its mass, whose co-ordinates are ∕, 0, h, are x ~l 7rp ≠ ⅛ + 2e '>- j μ∕' ⅛ c ≡) -1 ∕ c p° de J

o o q z = ∣ τp h j 1 y p d.c 3 (l + 2e)-^ J'pd(ed} + ^Jpde I

o θ q

We take into account the rotation of the earth by subtracting the centrifugal force ∕ω 2 = F from X. Now, the surface of constant density upon which the point f, 0, h is situated gives (1 - 2e)fdf + hdh = Q ∙, and the condition of equilibrium is that (X - F)<7∕^+ T i dh = 0. Therefore,

(X-F)Λ = Z∕(l-2e), which, neglecting small quantities of the order e 2 and putting ω¾ 2 = 4τr 2 , gives

*yi'<i.c>d+!⅛)-⅛√7⅛∙)-∣-∕⅛-⅛

0 0 <∙,

Here we must put now c for c 1 , c for r, and 1 + 2e under the first integral sign may be replaced by unity. Two integrations lead us to the following very important differential equation :—

d 2 e 2pc 2 de ∕ 2pc 6 ∖ de 1 fpc i dc de + ∖fpc 2 dc c i ) e ~^ θ ‘

When p is expressed in terms of c, this equation can be integrated. We infer then that a rotating spheroid of very small ellipticity, composed of fluid homogeneous strata such as we have specified, will be in equilibrium; and when the law of the density is expressed, the law of the corresponding ellipticities will follow. If we put M for the mass of the spheroid, then

M = y^∕*pd.c 3 (l + 2e); and

0

and putting c = c 0 in the equation expressing the condition of equilibrium, we find

M(2e -TO) = iτr.Ay r p rf( cc 5) f o

Making these substitutions in the expressions for the ∕ j 2 forces at the surface, and puttingr=1+6-e-, we get

Gcos<∕> = - ( l-e-^-m + (½m- 2e) ⅞- j — αc ∣ , 2 ∖ 2 ∕c 2 Jc

Gsinφ = - ( l + e~lm+(lm-2ey ’ L I A.

ac I 2 ∖ 2 ∕c 2 Jc

Here G is gravity in the latitude <∕>, and a the radius of

• C ∕ lb^ ∖

the equator. Since sec φ = y (l+e + e-^ ∖ , 0= “ { 1 4" ,+ (f~^ e ) siπiφ } ’ which expression contains the theorems we have referred to as discovered by Clairaut.

The theory of the figure of the earth as a rotating ellipsoid has proved an attractive subject to many of the greatest ma themati cans, Laplace especially, who has devoted a large portion of his Mécanique Celeste to it. In English the principal existing works on the subject are Sir George Airy’s Mathematical Tracts, where the subject is treated in the lucid style so characteristic of its author, but without the use of Laplace’s coefficients, Archdeacon Pratt’s Attractions and Figure of the Earth, and O’Brien’s Mathematical Tracts; in the last two Laplace’s coefficients are used. In the Cambridge Transactions, vol. viii., is a valuable essay by Professor Stokes, in which he proves, without making any assumption whatever as to the ellipticity of internal strata, or as to the past or the present fluidity of the earth that if the external form of the sea—imagined to percolate the land by canals—be a spheroid with small ellipticity, then the law of gravity will be that found above.

An important theorem by Jacobi must not be overlooked. He proved that for a homogeneous fluid in rotation a spheroid is not the only form of equilibrium; an ellipsoid rotating round its least axis may with certain proportions of the axes and a certain time of revolution be a form of equilibrium. Local Attraction.

In speaking of the figure of the earth, we mean the surface of the sea imagined to percolate the continents by canals. That this surface should turn out, after precise measurements, to be exactly an ellipsoid of revolution is a priori improbable. Although it may be highly probable that originally the earth was a fluid mass, yet in the cooling whereby the present crust has resulted, the actual solid surface has been left in form the most irregular. It is clear that these irregularities of the visible surface must be accompanied by irregularities in the mathematical figure of the earth, and when we consider the general surface of our globe, its irregular distribution of mountain masses, continents, with oceans and islands, we are prepared to admit that the earth may not be precisely any surface of revolution. Nevertheless, there must exist some spheroid which agrees very closely with the mathematical figure of the earth, and has the same axis of rotation. We must conceive this figure as exhibiting slight departures from the spheroid, the two surfaces cutting one another in various lines; thus a point of the surface is defined by its latitude, longitude, and its height above the spheroid of reference. Call this height for a moment η-, then of the actual magnitude of this quantity we can generally have no information, it only obtrudes itself on our notice by its variations. In the vicinity of mountains it may change sign in the space of a few miles; n being regarded as a function of the latitude and longitude, if its differential coefficient with respect to the former be zero at a certain point, the normals to the two surfaces then will lie in the prime vertical; if the differential coefficient of n with respect to the longitude be zero, the two normals will lie in the meridian; if both coefficients are zero, the normals will coincide. The comparisons of terrestrial measurements with the corresponding astronomical observations have ever been accompanied with discrepancies. Suppose A and B to be two trigonometrical stations, and that at A there is a disturbing force drawing the vertical through an angle δ, then it is evident that the apparent zenith of A will be really that of some other place A', whose distance from A is rδ, when r is the earth’s radius; and similarly if there be a disturbance at B of the amount δ', the apparent zenith of B will be really that of some other place B', whose distance from B is rδ'. Hence we have the discrepancy that, while the geodetical measurements deal with the points A and B, the astronomical observations belong to the points A', B'. Should δ, δ' be equal and parallel, the displacements AA', BB' will be equal and parallel, and no discrepancy will appear. The nonrecognition of this circumstance often led to much perplexity in the early history of geodesy. Suppose that, through the unknown variations of n, the probable error of an observed latitude (that is, the angle between the normal to the mathematical surface of the earth at the given point and that of the corresponding point on the spheroid of reference) be e, then if we compare two arcs of a degree each in mean latitudes, and near each other, say about five degrees of latitude apart, the probable error of the resulting value of the ellipticity will be approximately ± κ⅜σ<, « being expressed in seconds, so that if ∈ be so great as 2" the probable error of the resulting ellipticity will be greater than the ellipticity itself. It is not only interesting, but necessary at times, to calculate the attraction of a mountain, and the consequent disturbance of the astronomical zenith, at any point within its influence. The deflection of the plumb-line, caused by a local attraction whose amount is Aδ, is measured by the ratio of AS to the force of gravity at the station. Expressed in seconds, the deflection Δ is

Δ = 12".447.-A ,

P

where p is the mean density of the earth, δ that of the attracting mass,—the linear unit in expressing A being a mile. Suppose, for instance, a table-land whose form is a rectangle of twelve miles by eight miles, having a height of 500 feet and density half that of the earth; let the observer be two miles distant from the middle point of the longer side. The deflection then is Γ' ’ 472; but at one mile it increases to 2"∙20. At sixteen astronomical stations in the English Survey the disturbance of latitude due to the form of the ground has been computed, and the following will give an idea of the results. At six stations the deflection is under 2", at six others it is between 2" and 4'', and at four stations it exceeds 4". There is one very exceptional station on the north coast of Banffshire, near the village of Portsoy, at which the deflection amounts to 10'', so that if that village were placed on a map in a position to correspond with its astronomical latitude, it would be 1000 feet out of position! There is the sea to the north and an undulating country to the south, which, however, to a spectator at the station does not suggest any great disturbance of gravity. A somewhat rough estimate of the local attraction from external causes gives a maximum limit of 5", therefore we have 5" unaccounted for, or rather which must arise from unequal density in the underlying strata in the surrounding country. In order to throw light on this remarkable phenomenon, the latitudes of a number of stations between Nairn on the west, Fraserburgh on the east, and the Grampians on the south, were observed, and the local deflections determined. It is somewhat singular that the deflections diminish in all directions, not very regularly certainly, and most slowly in a south-west direction, finally disappearing, and leaving the maximum at the original station at Portsoy.

The method employed by Dr Hutton for computing the attraction of masses of ground is so simple and effectual that it can hardly be improved on. Let a horizontal plane pass through the given station; let r, θ be the polar coordinates of any point in this plane, and r, θ, z, the coordinates of a particle of the attracting mass; and let it be required to find the attraction of a portion of the mass contained between the horizontal planes 2 = 0, z — h, the cylindrical surfaces r = r 1 , r = r 2 , and the vertical planes θ≈θ l , θ = θ 2 . The component of the attraction at the station or origin along the line θ = 0 is s ∕* r2 ∕^¾ ∕* λ r 2 cos θdrclθdz J J J 0∙ 2 + z 2 )l n 0 ι o = bh (sin 0 2 - sin 0 1 ) log , r 2 + ^ 2 ⅛ , r 1 + (r 2 +Λ 2 )i'

By taking r 2 -r 1 sufficiently small, and supposing h also small, as it usually is, compared with r 1 + r 2 , the attraction is

7

=- δ(r 2 - r 1 )(sin 0 2 - sin θ 1 ) - , r where r = ⅛(r 1 + r 2 ). This form suggests the following procedure. Draw on the contoured map a series of equidistant circles, concentric with the station, intersected by radial lines so disposed that the sines of their azimuths are in arithmetical progression. Then, having estimated from the map the mean heights of the various compartments, the calculation is obvious.

In mountainous countries, as near the Alps and in the Caucasus, deflections have been observed to the amount of as much as 29". On the other hand, deflections have been observed in flat countries, such as that noted by Professor Schweitzer, who has shown that, at certain stations in the vicinity of Moscow, within a distance of 16 miles the plumb-line varies 16" in such a manner as to indicate a vast deficiency of matter in the underlying strata. But these are exceptional cases. Since the attraction of a mountain mass is expressed as a numerical multiple of δ: p, the ratio of the density of the mountain to that of the earth, if we have any independent means of ascertaining the amount of the deflection, we have at once the ratio p : δ, and thus we obtain the mean density of the earth, as, for instance, at Schiehallion, and more recently at Arthur’s Seat. A compact mass of great density at a small distance under the surface of the earth will produce an elevation of the mathematical surface which is expressed by the formula i (1 +⅛ s - ς λk cos 0)⅛ L· where a is the radius of the (spherical) earth, α(1 - Ä) the distance of the disturbing mass below the surface, ∕λ the ratio of the disturbing mass to the mass of the earth, and aθ the distance of any point on the surface from that point, say Q, which is vertically over the disturbing mass. The maximum value of y is at Q, where it is

The deflection at the distance aθ is δ _ μk sin θ

(1 j-fc 2 - 2k cos Θ)⅜ '

or since θ is small, putting h + lc— 1,

∆=- .

(A 2 + θ 2 )l

The maximum deflection takes place at a point whose distance from Q is to the depth of the mass as 1: 2, and its amount is

2 M 8√3Λ s '

Tf, for instance, the disturbing mass were a sphere a mile in diameter, the excess of its density above that of the surrounding country being equal to half the density of the earth, and the depth of its centre half a mile, the greatest deflection would be 5", and the greatest value of y only two inches. Thus a large disturbance of gravity may arise from an irregularity in the mathematical surface whose actual magnitude, as regards height at least, is extremely small.

The effect of the disturbing mass y on the vibrations of a pendulum would be a maximum at Q; if ν be the number of seconds of time gained per diem by the pendulum at Q, and σ the number of seconds of angle in the maximum deflection, then it may be shown that

V _ 7ΓV ’ 3 σ 1CΓ ’ so that the number of seconds of time by which at the maximum the pendulum is accelerated is about half the number of seconds of angle in the maximum deflection.

Principles of Calculation.

Let α, «' be the mutual azimuths of two points P, Q on a spheroid, k the chord line joining them, μ, μ' the angles made by the chord with the normals at P and Q, φ, φ', ω their latitudes and difference of longitude, and x + J- + ⅛ -1 = 0 the equation of the surface; α 2 o l then if the plane xz passes through P the co-ordinates of P and Q will be tX tx ’ *

X = — COS φ , » ≈ T7 cos Φ CO3 ω 9

Δ Δ

y=ο y ’ -^g cos <J> ’ sm « »

z=^(l-e 2 ) sin φ, z' =J^(l -e 2 ) sin φ , , where ∆ = (l -e s sin 2 ≠)i, ∆' = (1 -e 2 siu 2 φ ’ )f and e is the eccentricity. Let f, g, h be the direction cosines of the normal to that plane which contains the normal at P and the point Q, and whose inclinations to the meridian plane of P is = a ; let also ζ m, η and V, m ’ , n ’ be the direction cosines of the normal at P, and of the tangent to the surface at P which lies in the plane passing through Q, then since the first line is perpendicular to each of the other two and to the chord k, whose direction cosines are proportional to x'-x, y ’ -y, z ’ -z, we have these three equations

∕(a∕ -x) + gf + h(z ’- z) - 0 fl +gm +hn = 0 fl ’ + gm' + hn' — 0 .

Eliminate ∕, g l h from these equations, and substitute l = cos φ l'=- sin φ cos α

m = 0 m' = sin a n = sin φ n' = cos φ cos a, and we get

(a/- x) sin φ + y' cot a- <f-z) cos φ = 0 .

The substitution of the values of x, z, x', y , , z' in this equation will give immediately the value of cot «; and if we put £, C for the corresponding azimuths on a sphere, or on the supposition e — 0, the following relations exist cot α - cot C = e 2 c -^-, Q

cos φ Δ

cot a'- cot f'= -e 2 cos .^ ’- 51

cos φ Δ.

∆' sin φ-Δ sin φ , = sin ω Q .

If from Q we let fall a perpendicular on the meridian plane of P, and from P let fall a perpendicular on the meridian plane of Q, then the following equations become geometrically evident:

, . . ’ a » . k sin μ sin a = —, cos φ sin ω t~ δ z r k sin μ sin α = — cos φ sin ω .

Now in any surface u = 0 we have

⅛ 2 =(a∕-x) 2 + (y z -ι∕) 2 + (z , -z) 2

- COS μ - ∕⅛√∙ i k ∖ dz i+ df + dz i ) c03μ k ( da Ι + du ^ + d l ∖ i ∖ dz ’ * df i + dz ,2 7

in the present case, if we put xx' zz ’ _ τ - α i " F ’ then i cos μ=^∆U; cos∕Z=^-∆'U,

Let u be such an angle that

(1 - e 2 )⅛ sin φ = ∆ sin u cos φ = ∆ cos u, then on expressing x, x', z, z' in terms of u and u',

U = 1 ~ cos u cos «' cos « - sin u sin u!; also, if V be the third side of a spherical triangle, of which two sides are ∣π - u and ⅛π - u ’ and the included angle a>, using ‰ subsidiary angle ψ such that . , V u' -u u ’ + u sin ψ sin — = e sin cos —-— , τ 2 2 2 ’ we obtain finally the following equations :—

k = 2α cos ψ sin -

r 2

cos μ = Δ sec ψ sin ∣∙

cos μ —Λ sec ψ sm —

sin μ sin a = c L cos u , sin ω sin √ sin a = cos u sin « .

These determine rigorously the distance, and the mutual zenith distances and azimuths, of any two points on a spheroid whose latitudes and difference of longitude are given.

By a series of reductions from the equations containing f, it may be shown that z> 4

α + α'=f + ζ' + — ω(φ'-φ) 2 cos 4 φ 0 sin <∕> 0 + ....

4

where φ 0 is the mean of φ and φ', and the higher powers of e are neglected. A short computation will show that the small quantity on the right-hand side of this equation can never amount even to the ten thousandth part of a second, which is, practically speaking, zero; consequently the sum of the azimuths a + a on the spheroid is equal to the sum of the spherical azimuths, whence follows this very important theorem (known as Dalby’s theorem). If φ, φ' be the latitudes of two points on the surface of a spheroid, ω their difference of longitude, a, a their reciprocal azimuths, φ ’ - φ cos^- 5 - ι - , ω 2 a + a t, T . ,-+» cot - 2 ~∙

2

The vertical plane at P passing through Q and the vertical plane at Q passing through P cut the surface of the spheroid in two distinct curves. The greatest distance apart of these curves is, if α 0 = the mean azimuth of PQ, e 2 s 3

— - s cos 2 φ 0 sin 2α 0 .

This is a very small quantity; for even in the case of a line of 100 miles in length having a mean azimuth α= 45° in the latitude of Great Britain, it will only amount to half an inch, whilst for a line of fifty miles it cannot exceed the sixteenth part of an inch. The geodesic line joining P and Q lies wholly between these two curves. If we designate by P', Q' the two curves (the former being that in the vertical plane through P), then, neglecting quantities of the order e 2 0 3 , where θ is the angular distance of P and Q at the centre of the earth, the geodesic curve makes with P' at P an angle equal to the angle it makes with Q ’ at Q, each of these angles being a . third of the angle of intersection of P ’ and Q'. The difference of length of the geodesic line and either of the curves P', Q' is, s being the length of either, e 4 0 4 cos 4 φ 0 sin 2 2α u .

At least this is an approximate expression. Supposing the angle PQ to be as much as 10°, this quantity would be less than one hundredth of au inch.

An idea of the course of a geodesic line may be gathered from the following example. Let the line be that joining Cadiz and St Petersburg, whose approximate positions are

Cadiz. St Petersburg.

Lat. 36“ 22' N 59“ 56' N.

Long. 6° 18' w 30“ 17' e.

If G be the point on the geodesic corresponding to F on that one of the plane curves which contains the normal at

Cadiz (by “corresponding” we mean that F and G are on a meridian) then G is to the north of F; at a quarter of the whole distance from Cadiz GF is 458 feet, at half the distance it is 637 feet, and at three quarters it is 473 feet. The azimuth of the geodesic at Cadiz differs 20" from that of the vertical plane, which is the astronomical azimuth. The azimuth of a geodesic line cannot be observed, so that the line does not enter of necessity into practical geodesy, although many formjulae connected with its use are of great simplicity and elegance. The geodesic line has always held a more important place in the science of geodesy among the mathematicians of France, Germany, and Russia than has been assigned to it in the operations of the English and Indian triangulations. Although the observed angles of a triangulation are not geodesic angles, yet in the calculation of the distance and reciprocal bearings of two points which are far apart, and are connected by a long chain of triangles, we may fall upon the geodesic line in this manner :—

If A, Z be the points, then to start the calculation from A, we obtain by some preliminary calculation the approximate azimuth of Z, or the angle made by the direction of Z with the side AB or AC of the first triangle. Let P 1 be the point where this line intersects BC; then, to find P,, where the line cuts the next triangle side CD, we make the angle BP 1 P 2 such that BP 1 P 2 + BP 1 A = 180 o . This fixes P 2 , and P 3 is fixed by a repetition of the same process; soforP 4 , P 8 . . . · Now it is clear that the points P 1 , P 2 , P 3 so computed are those which would be actually fixed by an observer with a theodolite, proceeding in the following manner. Having set the instrument up at A, and turned the telescope in the direction of the computed bearing, an assistant places a mark P 1 on the line BC, adjusting it till bisected by the cross-hairs of the telescope at A. The theodolite is then placed over P 1 , and the telescope turned to A; the horizontal circle is then moved through 180°. The assistant then places a mark P 2 on the line CD, so as to be bisected by the telescope, which is then moved to P 2 , and in the same manner P 3 is fixed. Now it is clear that the series of points P 1 , P 2 , P 3 approaches to the geodesic line, for the plane of any two consecutive elements P n -ι P n , P n P«+i contains the normal at P n .

From the formulae which we have given above, expressing the mutual relations of two points P, Q on a spheroid, we may obtain the following solution of the problem: Given the latitude <∕> of P, with the azimuth α and distance s of Q, to determine the latitude and longitude of Q and the back azimuth a.

Let β = -∆

a e lQH

C= TΓλ κ cos * 2 ψ sin 2α

4(1 -e 2 )

⅝∙f⅛'' ,s,vrιi

£, ζ' are always veryminute quantities even for the longest distances; then, putting κ = 90 o - φ, tan <+<- ft, . si " ⅜( ,t - g -Ω cot “

2 sm⅜(κ + 0 + O 2

tan ≤+i±g g cos ⅜(<- fl -^ cot “

2 cos ⅜(κ + θ-r C) 2

φ -< t >= l. sin⅜(√+<- g ) ∕ √- g ∖ .

ψ ψ psin⅛(α'+^ +β )V 12 2 )’

here p is the radius of curvature of the meridian for the mean latitude ⅜(φ + φ'). These formulae are approximate only, but they are sufficiently precise even for very long distances.

Meridian Arcs.

The length of the arc of meridian between the latitudes φ 1 and φ 2 is s = f φ * p d φ = a

J Ρφ J (1-e 2 sin 2 <∕>)⅜ ’

Φι Φι

instead of using the excentricity, put the ratio of the axes = 1 - η: 1 + n , then s ∕'Φ2 ⅞(l+nχi-⅞ 2 )

f (1 + 2n cos 2φ + n a )⅜

Φl

This, after integration, gives

5 = (1 + n + ( ¼ ∣ ( 3» + 3»> + ?‘»s) «, + (£»’+ ⅛⅛) α , where α 0 = ≠2 - Φl a 1 = sin (φι~φ 1 ) cos (≠ 2 + ≠ι) a 2 = sin 2(ψ 2 - ’ ≠ 1 ) cos 2(<f> 2 ÷φ 1 ) a 3 = sin 3(<⅛> 2 -<j> 1 ) cos 3(< ∣ ⅛ 2 +φ 1 ) The part of s which depends on n 3 is very small; in fact, if we calculate it for the longest arc measured, the Bussian arc, it amounts to only an inch and a half, therefore we omit this term, and put for ⅜ the value o

(1 +n + ∣ n 2 ) α 0 - (3n + 3?i 2 ) a 1 + (^n 2 )a 2

Now, if we suppose the observed latitudes to be affected with errors, and that the true latitudes are φ 1 + x 1 , φ 2 + x 2 ; and if further we suppose that n i + dn is the true value of a-b: a + b, and that n 1 itself is merely a very approximate numerical value, we get, on making these substitutions and neglecting the influence of the corrections x on the position of the arc in latitude, i.e., on φ 1 + φ 2 ,

I - (1 + n 1 + I ¾) α 0 - ( 3n 1 +3nJ) α 1 + (^nj)α 2 + {(l+ ∣ n ι) a o - ( 3 + 6,ι ι) a ι+(ι ’ h)<⅛ } dn + I l+n 1 -3w 1 ⅛pa 0 ; here da^~x 2 - x 1 ; and as b is only known approximately, put b 1 - b{l + u)∙, then we get, after dividing through by the coefficient of da 0 , which is = 1 + π 1 - 3n 1 cos (φ 2 - <f> 1 ) cos (φ 4 - φ 1 ), an equation of the form x 2 =x 1 + h+fu + gυ, where for convenience we put v for dn.

Now in every measured arc there are not only the extreme stations determined in latitude, but also a number of intermediate stations, so that if there be i +1 stations there will be i equations

α⅛=a⅛+∕ 1 tt+gqv + A 1 ¾ = ≈1+Λ w + ^ + λ 2 Xi = X i +fiU + giv + hi .

In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each arc will supply a number of equations in u and v and the corrections to its observed latitudes. Then, according to the method of least squares, those values of u and υ are the most probable which render the sum of the squares of all the errors x a minimum. The corrections x which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modern instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections x to the observed latitudes are attributable to local attraction. Archdeacon Pratt, in his treatise on the figure of the earth, objects to this mode of applying least squares first used by Bessel; but certainly Bessel was right, and the objection is groundless.

Comparisons of Standards.

In determining the figure of the earth from the arcs of meridian measured in different countries, one source of uncertainty was, until the last few years, the want of comparisons between the standards of length in which the arcs were expressed. This has been removed by the very extensive series of comparisons recently made at Southampton (see Comparisons of Standard of Length oj England, France, Belgium, Prussia, Russia, India, and Australia, made at the Ordnance Survey Office, Southampton, 1866, and a paper in the Philosophical Transactions for 1873, by Lieut.-Col. A. R. Clarke, C.B., R.E., on the further comparisons of the standards of Austria, Spain, the United States, Cape of Good Hope, and Russia). These direct comparisons, which were carried out with the highest attainable precision, are of very great value. The length of the toise has three independent determinations, viz., through the Russian standard double toise, the Prussian toise, and the Belgium toise,—giving for the length of the toise, expressed in terms of the standard yard of England through the Russian standard 6 39453216 ft.

„ ,, Prussian ,, . 6 39453703 ft.

„ „ Belgian „ 6 39453215 ft.

By combining all the different comparisons made in England and on the Continent on these bars, by the method of least squares, the final value of the toise is 6∙39453348 ft (log = 0‘8058088656), from which the greatest divergence of the three separate results specified above is only half a millionth of a toise, corresponding to ten feet in the earth’s radius. From the known ratio of the toise and the metre, 864000: 443296, we get for the metre

3∙28086933 ft. (log = 0'5159889356).

That the close agreement between the determinations of the toise is not due to chance will be seen from the fact that the comparisons of the Prussian toise with the English standard involved 2340 micrometer readings and 520 thermometer readings, extending over twenty-five days, the probable error of the resulting length of the toise being ±0'00000015 yard. The probable error of the determination of the Belgian toise is ÷ 0Ό0000027; that of the Russian double toise ±0O0000031. With regard to tne metre, there is an independent determination resulting from the comparison of the platinum metre of the Royal Society,— a large number of observations giving for the length of the metre 3 ’ 28087206 feet, which differs from the former result by about one millionth part. But this determination, involving the expansion of the bar for 30° of temperature, and being dependent on some old observations of Arago, cannot be allowed any weight in modifying the result obtained through the toises. The Russian standard, compared at Southampton, was that on which the length of their base lines and therefore their whole arc depends.

Calculation of the Semiaxes.

We now bring together the results of the various meridian arcs, omitting many short arcs which have been used in previous determinations, but which on account of their smallness have little influence in the result aimed at.

The data of the French arc from Formentera to Dunkirk are— [table]

The latitude of Formentera as here given is taken from the observations of Μ. Biot, recorded and computed in the third volume of his Traité Élémentaire d ’ Astronomie physique.

The latitude of the Pantheon, given in the Base du Système Métrique Decimal (ii. 413), is 48° 50' 48"∙86. In the Annales de l' Observatoire Imperial de Paris, vol. viii. page 317, we find the latitude of south face of the observatory determined as 48° 50' 11"‘71. The Pantheon being 35" - 38 north of this, we thus get a second determination of its latitude. The mean is that given above.

The distance of the parallels of Dunkirk and Greenwich, deduced from the recent extension of the triangulation of England into France, in 1862, is 161407 , 3 feet, which is 3 - 9 feet greater than that obtained from Captain Kater’s triangulation, and 3'2 feet less than the distance calculated by Delambre from General Roy’s triangulation. The following table shows the data of the English arc with the distances in standard feet from Formentera. [table]

The latitude assigned in this table to Saxavord is not the directly observed latitude, which is 60° 49' 38" ’ 58, for there are here a cluster of three points, whose latitudes are astronomically determined; and if we transfer, by means of the geodesic connection, the latitude of Gerlh of Scaw to Saxavord, we get 60° 49' 36" ’ 59; and if we similarly transfer the latitude of Balta, we get 60° 49' 36" ’ 46. The mean of these three is that entered in the above table.

For the Indian arc in long. 77° 40' we have the following data :— [table]

The data of the Russian arc (long. 26° 40') taken from M. Struve’s work are as below :— [table]

From the arc measured by Sir Thomas Maclear in long. 18° 30', we have [table]

And, finally, for the Peruvian arc, in long. 281° 0', [table]

Having now stated the data of the problem, we may either seek that ellipsoid which best represents the observations, or we may restrict the figure to one of revolution. It will be convenient to commence with the supposition of an ellipsoidal figure, as on so doing we can, by a slight alteration in the equations of minimum, obtain also the required figure of revolution. It may be remarked that, whatever the real figure may be, it is certain that if we presuppose it an ellipsoid, the arithmetical process will bring out an ellipsoid, which ellipsoid will agree better with all the observed latitudes than any spheroid would, therefore we do not prove that it is an ellipsoid; to prove this, arcs of longitude would be required. There is no doubt such arcs will be shortly forthcoming, but as yet they are not available.

The first thing that occurs to one in considering an ellipsoidal earth is the question, What is a meridian curve 1 It may be defined in different ways: a point moving on the surface in the direction astronomically determined as “north” might be said to trace a meridian; or we may define it as the locus of those points which have a constant longitude, whose zeniths lie in a great circle of the heavens, having its poles in the equator; we adopt this definition. Let a, b, c be the semiaxes, c being the polar semiaxis. The equation of the ellipsoid being

α 8 5≡ + c a ’

if P be any point on the surface, the direction cosines of the normal at P are proportional to du du (lu λt , « y_ 2 ∙ . dx’ Ty’ te’ α 2 ’ P ’ c 2 ’ and if ⅜π - φ be the angle between this normal and the •minor axis, so that φ is the latitude of P, we have z βin ≠ = ∕x 2 y 2 ⅛∖⅜' ∖α 4 6 4 c 4 √

Hence the equation to a “parallel” in which the latitude φ is constant is

⅞ + g-C∞t≡ i > = O. .

α 4 b i c i

So that in an ellipsoidal earth the parallel is no longer a plane curve. Let longitude be reckoned from the plane of xz. As there are two species of latitude, astronomical and geoeentric, so there are in the ellipsoidal earth two species of longitude, geocentric (called «) and astronomical (called ω). Conceive a line passing through the origin in the plane of the equator and directed to a point whose longitude is ⅛π + ω. The direction cosines of that line are—sin ω, cos ω, and 0. Those points of the surface whose normals are at right angles to this line are in the meridian whose longitude is ω; the condition of perpendicularity is expressed by X ein ω y cos ω _ n ~aΓ~ +- P ’

and this, in fact, is the equation of the meridian, which is still on the ellipsoidal hypothesis a plane curve. The geocentric and astronomical longitudes are connected by the relation

α 2 tan u= b i tan a>.

This meridian curve is an ellipse whose minor semi-axis is c, and of which the semi-axis major is some quantity r intermediate between a and b, such that

1 _ cos 2 u sin 2 u r 2 α 2 ~ ~⅛ 2 ~ *

Take two quantities i, k, such that α 2 (l -i) = δ 2 (l + i')-k 2 , then ⅛ 2 =r 2 (l - i cos 2w); and take n such that

„_r-c

-—- , r + e and substitute the value of r, neglecting the square of i ; this gives

»“V—- + v cos 2ω .

k + c 4

Now we have to determine not only the three semi-axes a, b, e, but the longitude of a. Let w 1 be the longitude of one of the measured meridian arcs, u 0 the longitude of a, then, for that arc,

⅛-c i n , λ n = *Tc + 4 C0S =JHM-+p cos 2w j +o sin 2u 1 , k + c where 4p = i cos 2u 0 , 4q-i sin 2w 0 .

The normal at P does not pass through the axis of rotation, so that the observed latitudes on an ellipsoid are not exactly the quantities which should be used in the ordinary method of expressing the length of a meridian arc in terms of the latitudes. But it may be shown that this consideration may be neglected.

The data we have collected form 35 equations between the 40 «-corrections to the observed latitudes, and the four unknown quantities determining the elements of the ellipsoid. Suppose n 1 to be an approximate value of the ratio fc - c: lc + c, so that k-c vrr" ι+r ∙

where r is a small correction to n 1 and suppose c 1 to be an approximate value of c so that c = c 1 ( 1 + t), then the four unknown quantities are p, q, r, t. The result of making the sum of the squares of the 40 corrections a minimum is

Feet. Metres.

α =-∙ 20926350 = 6378294 0 b = 20919972 = 6376350'4 c =20853429 = 6356068’1

a — c 1 6—c 1

^^c~ 285 - 97 ’ ~c ' - 313∙38

a — b J

- c -- 3269 τ 5 Longitude of a 15° 34' East.

The meridian of the greater axis passes, in the Eastern Hemisphere, through Spitzbergen, the Straits of Messina, Lake Chad in North Africa, and along the west coast of South Africa,—nearly corresponding to the meridian which passes over the greatest quantity of land in that hemisphere. In the Western Hemisphere it passes through Behring’s Straits and through the centre of the Pacific Ocean. The meridian (105° 34' E.) of the minor axis of the equator passes near North-east Cape on the Arctic Sea, through Tong-king and the Straits of Sunda, and corresponds nearly to the meridian which passes over the greatest amount of land in Asia; and in the Western Hemisphere it passes through Smith Sound, the west of Labrador, Montreal, between Cuba and Hayti, and along the west coast of South America, nearly coinciding with the meridian that passes over the greatest amount of land in that hemisphere.

The length of the meridian quadrant passing through Paris, in the ellipsoidal figure given above, is 10001472 ’ 5 metres, showing that the length of the ideal French standard is considerably in error as representing the ten-millionth part of the quadrant. The minimum quadrant, in longitude 105° 34', has a length of 10000024 ’ 5 metres. The probable error of the longitude of the major axis of the equator given above is of course large, as much perhaps as ±15°.

It has been objected to this figure of three unequal axes that it does not satisfy, in the proportions of the axes, the conditions brought out in Jacobi’s theorem. Admitting this, it has to be noted, on the other hand, that Jacobi’s theorem contemplates a homogeneous fluid, and this is certainly far from the actual condition of our globe, indeed, the irregular distribution of continents and oceans suggests as possible a sensible divergence from a perfect surface of revolution.

If we limit the figure to being an ellipsoid of revolution, we get rid in our equations of two unknown quantities, and the result may be expressed thus :—

Feet. Metres.

α = 20926062 = 6378206∙4

c =20855121 = 6356503 8 c: α = 293 ’ 98 :294 - 98.

As might be expected, the sum of the squares of the 40 latitude corrections, viz., 153’99, is greater in this figure than in that of three axes, where it amounts to 138 ’ 30. In the Indian arc the largest corrections are at Dodagoontah, + 3"∙87, and at Kalianpur, -3" ’ 68. In the Russian arc the largest corrections are + 3" ’ 76, at Tornea, and - 3" - 31, at Staro Nekrassowka. Of the whole 40 corrections, 16 are under 1"O,10 between 1" ’ 0 and 2".0, 10between 2"O and 3"O, and 4 over 3"O. For the ellipsoidal figure the probable error of an observed latitude is ± 1" ’ 42; for the spheroidal it would be very slightly larger. This quantity may be taken therefore as approximately the probable amount of local deflection.

In 1860, the Russian Government, at the instance of M. Otto Struve, imperial astronomer at St Petersburg, invited the co-operation of the Governments of Prussia, Belgium, France, and England, to the important end of connecting their respective triangulations so as to form a continuous chain under the parallel of 52° from the island of Valentia on the south-west coast of Ireland, in longitude 10° 20' 40" W., to Orsk on the river Ural in Russia. This grand undertaking was at once set in action, but up to the present time there are portions of the work still incomplete. On the part of England the triangulation was, in 1862, carried through France into Belgium; and the difference of longitude of Greenwich and Valentia was determined by the Astronomer Royal by means of electric telegraph signals.

Although in theory the determination of differences of longitude by electric telegraph signals may appear ext remely simple, yet practically there are very many sources of error which have to be sought out and eliminated by a proper arrangement of the observations. The system has now been brought to such perfection that the astronomical amplitude of arcs of longitude can be determined with nearly as much accuracy as those of latitude, and in a few years the data of the problem of the figure of the earth will thus receive many additions. As an example of the precision arrived at, the difference of longitude of Greenwich Observatory and Harvard Observatory, U.S.A., has been three times determined with the following results :— [table]

But the different determinations of the velocity of transmission of signals present great anomalies.

Pendulum Observations.

In Clairaut’s theorem we have seen that if f be gravity in the latitude of <∕>, g its value at the equator, then ∕ = <7(1+ <7 sin 2 φ). If the same pendulum be swung in different latitudes then the square of the number of vibrations will be proportional to gravity. Hence, if N be the number of vibrations of an invariable pendulum per diem at the equator, N' the number in latitude <∕>, then N' 2 = N 2 (1 + q sin 2 φ). Thus q can be obtained by observations on the same pendulum in different latitudes, and since q — ^m-e and m is known, e will at once follow. The pendulum which makes 86400 oscillations per diem in London is observed to lose 136 vibrations at the equator and gain 79 at Spitzbergen.

The limits of space at our disposal here prevent our going into the subject of pendulum experiments, and it seems unnecessary to repeat the investigations that have already been based upon the older pendulum observations. See Airy’s Figure of the Earth, Baily’s paper in the Memoirs of the Royal Astronomical Society, General Sabine’s Account of Experiments to determine the Figure of the Earth by means of the Pendulum vibrating seconds in Different Latitudes, 1825, and a valuable paper in the Cambridge Philosophical Transactions, 1849, by Professor Stokes. The pendulum gives an ellipticity certainly somewhat greater than that resulting from arcs of meridian, viz., - 2 -g⅛.- s -. An immense number of pendulum observations are now being made at the astronomical stations of geodesical surveys in Germany, Russia, and India, which, when fully published, will throw light more perhaps upon the local variations of gravity than on the figure of the earth. The observations made at the various stations of the Indian meridian arc bring to light a physical fact of the very highest importance and interest, namely, that the density of the strata of the earth’s crust under and in the vicinity of the Himalayan Mountains is less than that under the plains to the south, the deficiency increasing as the stations of observation approach the Himalayas, and being a maximum when they are situated on the range itself. This accounts fer the non-appearance of the large deflections which the Himalayas, according to Archdeacon Pratt’s calculations, ought to produce. The Indian pendulum observations also throw some light on the relative variations of gravity at continental, coast, and island stations, showing that, without a single exception, gravity at the coast stations is greater than at the corresponding continental stations, and greater at island stations than at coast stations. The ellipticity of the earth has also been deduced from the motion of the moon, the quantity e-⅛m entering as a coefficient_in the expression for the moon’s latitude. The resulting value of the ellipticity is ∙ 2 -⅜ τ th (Airy’s Tracts, p. 188). A value of the ellipticity may also be derived from the precession of the equinoxes, but as this depends on the assumed law of density in the interior of the earth it is not of much importance.

Elements of the Figure as a Solid of Revolution. α=20926062: δ=20855121 .

If p be the radius of curvature of the meridian in latitude φ, p' that perpendicular to the meridian, D the length of a degree of the meridian, D' the length of a degree of longitude, r the radius drawn from the centre of the earth, V the angle of the vertical, then [table]