BALANCE. For the measurement of the “mass” of (i.e., of the quantity of matter contained in) a given body we possess only one method, which, being independent of any supposition regarding the nature of the matter to be measured, is of perfectly general applicability. The method—to give it at once in its customary form—consists in this, that after having fixed upon a unit mass, and procured a sufficiently complete set of bodies representing each a known number of mass-units (a “set of weights ”), we determine the ratio of the weight of the body under examination to the weight of the unit piece of the set, and identify this ratio with the ratio of the masses. Machines constructed for this particular modus of weighing are called balances. Evidently the weight of a body as determined by means of a balance—and it is in this sense that the term is always used in everyday life, and also in certain sciences, as, for instance, in chemistry—is independent of the magnitude of the force of gravity; what the merchant (or chemist) calls, say, a “pound” of gold is the same at the bottom as it is at the top of Mont Blanc, although its real weight, i.e., the force with which it tends to fall, is greater in the former than it is in the latter case.
To any person acquainted with the elements of mechanics, numerous ideal contrivances for ascertaining which of two bodies is the heavier, and for even determining the ratio of their weights, will readily suggest themselves; but there would be no use in our noticing any of these many conceivable balances, except those which have been actually realised and successfully employed. These may be conveniently arranged under six heads.
1. Spring Balances. — The general principle of this class of balances is that when an elastic body is acted upon by a weight suspended from it, it undergoes a change of form, which, caeteris paribus, is the greater the greater the weight. The simplest form of the spring balance is a straight spiral of hard steel (or other kind of elastic) wire, suspended by its upper end from a fixed point, and having its lower end bent into a hook, from which, by means of another hook crossing the first, the body to be weighed is suspended,— matters being arranged so that even in the empty instrument the axis of the spiral is a plumb-line. Supposing a body to be suspended at the lower hook, it is clear that the point where the hooks intersect each other will descend from the level it originally occupied, and that it must fall through a certain height h before it can, by itself, remain at rest. This height, provided the spiral was not strained beyond its limit of elasticity {i.e., into a permanent change of form), is proportional to the weight P of the body, and consequently has to the mass Μ the relation Λ = cyM, where c is a constant and g the acceleration of gravity. Hence, supposing in a first case h and Μ to have been h' and M', and in a second case, h!' and M*, we have h': h" :: <7′M′: f'W; and it is only as long as g is the same that we can say K : h! ’ ∙. : M': Μ*. Spring balances are very extensively used for the weighing of the cheaper articles of commerce and other purposes, where a high degree of precision is not required. In this class of instruments, to combine compactness with relatively considerable range, the spring is generally made rather strong; and sometimes the exactitude of the reading is increased by inserting, between the index and that point the displacement of which serves to measure the weight, a system of levers or toothed wheels, constructed so as to magnify into convenient visibility the displacement corresponding to the least difference of weight to be determined. Attempts to convert the spring balance into a precision instrument have scarcely ever been made; the only case in point known to the writer is that of an elegant little instrument constructed by Professor Jolly, of Munich, for the determination of the specific gravity of solids by immersion, which consists of a long steel-wire spiral, suspended in front of a vertical strip of silvered glass bearing a millimetre scale. To read off the position of equilibrium of the index on the scale, the observing eye is placed in such a position that the eye, its image in the glass, and the index are in a line, and the point on the scale noted down with which the index apparently coincides.
2. Chain Balances. — This invention of Wilhelm Weber’s having never, so far as we know, found its way into actual practice, we confine ourselves to an illustration of its principle. Imagine a flexible string to have its two ends attached to the two fixed points C and D (fig. 1), forming the terminal points of a horizontal line CD shorter than the string. Suppose two weights to be suspended, the one at a point A, the other at a point B of the string; the form of the polygon CDBA will depend, caeteris paribus, on the ratio of the two weights. Assuming, for simplicity’s sake, CA to be equal to DB, then, if the weights are equal, say, each = P units, the line AB will be horizontal. But if now, say, the weight at B be replaced by a heavier weight Q, the point A will ascend through a height h, the point B will descend through a lesser height h' in accordance with equation PΛ = QΛ', and the angle between what is now the position of rest of the base line A'B', and the original line AB will depend on the ratio of P: Q. The exact measurement of this angle would be difficult, but it would be easy to devise very exact means for ascertaining whether or not it was horizontal, and, if not, whether it slanted down the one way or the other; and thus the instrument might serve to determine whether P was equal to, or greater or less than, Q; and this obviously is all that is required to convert the contrivance into an exact balance.
3. Lever Balances. — This class of balances, being more extensively used than any other, forms the most important division of our subject. There is a great variety of lever balances; but they are all founded upon the same principles, and it is consequently expedient to begin by summing up these into one general theory.
Theory of the Lever Balance (fig. 2).—In developing the “theory” of a machine, the first step always is and must be that we substitute for the machine as it is a fictitious machine, which, while it closely corresponds in its working to the actual thing, is free from its defects. In this sense what now follows has to be understood. Imagine an inflexible beam suspended from a stand in such a manner that, while it can rotate freely about a certain horizontal axis fixed in its position with respect to both the stand and the beam, and passing through the latter somewhere above its centre of gravity, it cannot perform any other motion. Imagine the beam at each end to be provided with a vertical slit, and each slit to be traversed by a rigid line fixed in the beam in such a situation that both lines are parallel to, and in one and the same plane with, the axis of rotation; and suppose the mass of the beam to be so distributed that the line connecting the centre of gravity S with its projection 0 on the axis of rotation stands perpendicular on that plane. Suppose now two weights, P' and P", to be suspended by means of absolutely flexible strings, the former from a point A on the rigid line in the left, the other from a point B on the rigid line in the right slit, and clearly, whatever may be the effect, it will not depend on the length of the strings. Hence we may replace the two weights by two material points situated in A and B, and weighing P' and P* respectively. But two such points are equivalent, statically, to one point (weighing P' + P") situated somewhere in D within the right line connecting A with B. Suppose the beam to be arrested in its “normal position” (by which we mean that position in which AB stands horizontal and the line SO is a plumb-line), and then to be released, the statical effect will depend on the situation of the point D, and this situation, supposing the ratio l': I" to be given, on the ratio P': P' ’. If P ’ l' = P" I", D lies in the axis of rotation; the beam remains at rest in its normal position, and, if brought out of it, will return to it, being in stable equilibrium. This at once suggests two modes of constructing the instrument and two corresponding methods of weighing.
First Method.— We so construct our instrument that while ΐ is constant, I" can be made to vary and its ratio to ΐ be measured. In order then to determine an unknown weight P', we suspend it at the point pivot A; we then take a standard weight P" and, by shifting it forwards and backwards on AB, find that particular position of the point of suspension B, at which P* exactly counterpoises P'. We I" I" then read off y , and have P' = P*y. But, practically, the body to be weighed cannot be directly suspended from A, but must be placed in a pan suspended from A, and consequently the weight p 0 of the pan and its appurtenances would always have to be deducted from the total weight P', as found by the experiment, to arrive at the weight of the object p ≈ P' - p 0 . Hence, what is actually done in practice is so to shape the right arm that its back coincides with the line AB, and to lay down on it a scale, the degrees of which are equal to one another, and to l' (or some convenient submultiple or multiple of l') in length, and so to adjust p 0 and number the scale, that when the sliding weight P" is suspended at the zero-point, it just counterpoises the pan; so that when now it is shifted successively to the points 1, 2, 3 . . . p, it balances exactly 1, 2, 3 . . . p units of weight placed in the pan. This is the principle of the common steel-yard, which, on account of the rapidity of its working, and as it requires only one standard weight, is very much used in practice for rough weighings, but which, when carefully constructed and adjusted, is susceptible of a very considerable degree of precision. In the case of a precision steel-yard, it is best so to distribute the mass of the beam that the right arm balances the left one + the pan, to divide that arm very exactly into, say, only 10 equal parts, and instead of one sliding weight of P" units to use a set of standards weighing P ,, , τ ⅛ P", τ ⅛ σ P", 10 1 0 σ P", &c. The great difficulty is to ensure to the heavier sliding weights a sufficiently constant position on the beam. To show the extent to which this difficulty can be overcome it may be stated that in an elegant little steel-yard, constructed by Mr Westphal of Celle (for the determination of specific gravities), which we had lately occasion to examine, even the largest rider, which weighs about 10 grammes, was so constant in its indications that, when suspended in any notch, it always produced the same effect to within less than 3 0 1 0 - σ th of its value.
Second Method.— We so construct our instrument that both l' and Γ have constant values, and are nearly or exactly equal to each other, and provide it with pans, whose weights p 0 ' and pf are so adjusted against each other that p' l' = pfl', and, consequently, the empty instrument is at rest in its normal position. We next procure a sufficiently complete set of weights, i.e., a set which, by properly combining the several pieces with one another, enables us to build up any integral multiple of the smallest difference of weight δ we care to determine, a set, for instance, which virtually contains any term of the series 0Ό01, 0Ό02, 0.003 100'000 grammes. In order now to determine an unknown weight p', we place it, say, in the left pan, and then, by a series of trials, find that combination of standards p>' which, when placed in the right pan, establishes equilibrium to within ± δ. Evidently— ∕-fci≡ d).
In the case of purely relative weighings, there is nothing to
Γ ∕ I" ∖ hinder us from adopting y units (e.g., y grammes! as our unit of mass, and simply to identify the relative value of p' with the number p'. But even if we want to know the absolute value of p in true grammes, we need not know the numerical value of —. AU we have to do is, after
Γ ’ having determined the value of p' in terms of y, to reverse the positions of object and standards, and, in a similar manner, to ascertain the value p 1 " which now counterpoises the unknown weight p' lying in the right pan. Obviously p' ^P" γ= Pi" (pΥ≈p"pι'∙> and P' = *Jp v p"> for which expression, if the two arms are very nearly of equal length, we may safely substitute p' = ⅛(p" +p 1 "). Or, instead of at once finding the counterpoise for p' in standards, we may first counterpoise it by means of shot or other material placed in the opposite pan, and then find out the number of grammes p" which has to be substituted for p' to again establish absolute equilibrium. Evidently p' = p". This (in reference to the ideal machine meant to be realised) is the theory of the common balance as we see it working in every grocer’s shop, and also that of the modern precision balance, which, in fact, is nothing but an equal-armed beam and scales refinedly constructed. In the case of the latter class of balances the inconvenience involved in the use of very small weights may be avoided (and is generally avoided) by dividing the right arm of the beam, or rather the line AB, into 10 equal parts, and determining differences of less than, say, 0Ό1 gramme by means of a sliding weight possessing that value. But evidently, instead of dividing the whole length of the right arm, it is better to divide some portion of it which is so situated that the rider can be shifted from the very zero to the “10,” and so to adjust the rider, that when it is shifted successively from 0 to 1, 2, 3 . . . n it is the same as if 1, 2, 3 . . . n tenths of its weight were placed in the right pan. The rider in this case must, of course, form part and parcel of the beam. It is singular that none of our precision-balance makers have ever thought of this very obvious improvement on the customary system. In the very excellent instrument made by Messrs Becker and Company of New York, this, it is true, is realised partially in a rider weighing 12 milligrammes and a beam divided into 12 equal parts (instead of 10 and 10 respectively); but this does not enable one to shift the rider to where it would indicate from 0 to say -⅛ or 1 ⅛∙ of a milligramme. Whichever of these modes of weighing we may adopt, we must have an arrangement to see whether the balance is in its normal position, and it is desirable also to have some means to enable us, in the course of our trials, to form at least an idea as to the additional weight which would have to be added to the standards on the pan (or to be taken away) in order to establish equilibrium. To define the normal position, all that is required is to provide the beam with a sufficiently long “needle,” the axis of which is parallel to the line OS, and which plays against a circular limb fixed to the stand and constructed so that the upper edge of the limb coincides very nearly with rhe path of the point of the vibrating needle, and to graduate the limb so that, as fig. 2 shows, the zero point indicates the normal position of the beam. In order to see how the graduation must be made to be as convenient as possible a means for translating deviations of the needle into differences of weight, let us assume the balance to be charged with P' grammes from A and with P" + Δ grammes from B, and P' and P" to satisfy the equation P' l' = P"Z ’ '. The two weights P' and P' being equivalent to one point P' + P" in the axis of rotation, the effect is the same as if these two weights did not exist and the beam was only under the influence of two weights, viz., the weight W of the beam acting in S and the weight Δ acting in B. But this comes to the same as if both W and Δ were replaced by one point weighing W + Δ, and situated somewhere at C o between, and on a line with, B and S. Hence, supposing the beam to be first arrested in its normal position and then to be left to itself, the right arm will go down and not be able by itself to remain at rest before it has reached that position in which C o lies vertically below the axis of rotation. Coeteris paribus C o will be the nearer to B, and consequently the angle α, through which the beam (and with it the needle) has to turn to assume what now is its position of stable equilibrium, will be the greater the greater Δ is, and for the same Δ and W the angle of deviation will be the greater the less the distance s of the centre of gravity of the beam S is from the axis of rotation. The former proposition enables one in a given case to form an idea of the amount Δ which has to be taken away from the right pan to establish equilibrium. To find the exact mathematical relation between Δ and the corresponding angle α, let us remember that the position of C o is the same whatever may be the direction of gravity with regard to the beam. Assuming gravity to act parallel to OS, we have (W + Δ) CC 0 = ΔZ", where C stands for the projection of C o on OS. Assuming, secondly, gravity to act parallel to the line OB, we have (W + Δ). C0 = W. OS;
CC 0 ÷ ∕O<
CO Ws v '
Obviously, the right way of graduating the limb is to place the marks so that their radial projections on the tangent to the circle at the zero-point divide that line into parts of equal length. In the Ordinary balance where Z" is a constant, the factor — has a constant value, which can be determined by one experiment with a known Δ—always supposing that in the instrument used the requirements of our theory were exactly fulfilled. In good precision balances they are fulfilled, to such an extent at least, that although the factor named is not absolutely constant, but a function of P, it can be looked upon as a relative constant, so that by determining the deviations produced by a given Δ, say Δ = 1 milligramme, for a series of charges (z.e., values of P"), one is enabled to readily convert deviations of the needle, as read off on the scale, into differences of weight. This method is very generally followed in the exact determinations of weights as required in chemical assaying, in the adjusting of sets of weights, &c. Only, instead of letting the needle come to rest and then reading off its position, what is done is to note down 2, 3, 4 . . . n consecutive excursions of the needle, and from the readings (α 1 , a i , α 3 , a i . . . α β ) to calculate the position a 0 where the needle would come to rest if it were allowed to do so. It being understood that the readings must be taken as positive or negative quantities according as they lie to the left or to the right of the zero-point, α 0 might be identified with any of the sums—
⅛ (α 1 + α 2 ), I (α 2 + α 3 ), ⅛ («.-i + α n ), but clearly it is much better to calculate α 0 by taking the mean of these quantities, thus—
α 1 ÷α.+2 (c⅛+a 3 ... + <⅛-ι).
β ° - - 2(n-l) 5 and it is also easily seen that to eliminate as much as possible the influences of the resistance of the air and
(let us at once add by anticipation of what ought to be reserved for a subsequent paragraph) of the friction in the pivots of the balance, it is expedient to let n be an odd number. Theoretically this method is. of course, not confined to small Δ⅛, and it is easy to conceive a balance in which the limb is so graduated that it gives directly the weight of an object placed in the right pan; this is the principle of the Tangent Balance, a class of instruments which used to be very generally employed for the weighing of letters, parcels, &c., but is now almost entirely superseded by the spring balance.
After having thus given a general theory of the ideal, let us now pass to the actual instrument. But in doing so we must confine ourselves mainly to the consideration of that particular class of instruments called precision balances, which are used in chemical assaying, for the adjustment of standard weights, and for other exact gravimetric work.
The Precision Balance being, as already said, quite identical in principle with the ordinary u pair of scales,” there is no sharp line of demarcation between it and what is usually called “a common balance,” and it is equally impossible to name the inventor of the more perfect form of the instrument. But taking the precision balance in what is now considered its most perfected form, we may safely say that all which distinguishes it from the common balance proper is, in the main, the invention of the late Mr Robinson of London. In Robinson’s, as in most modern precision balances, the beam consists of a perforated flat rhombus or isosceles triangle, made in one piece out of gun-metal or hard-hammered brass. The substitution for either of those materials of hard steel would greatly increase the relative inflexibility of the beam, but, unfortunately, steel is given to rusting, and, besides, is apt to become magnetic, and has therefore been almost entirely abandoned. The perforations in the beam are an important feature, as they considerably diminish its weight (as compared with what that would be if the perforations were filled up) without to any great extent reducing its relative solidity. In fact, the loss of carrying power which a solid rhombus suffers in consequence of the middle portions being cut out, is so slight that a very insignificant increase in the size of the minor diagonal is sufficient to compensate for it. Why a balance beam should be made as light as possible is easily seen; the object (and it is as well here to say at once, the only object) is to diminish the influence of the unavoidable imperfections of the central pivot. To reduce these imperfections to a minimum, the beam in all modern balances is supported on a polished horizontal plane of agate or hard steel fixed to the stand, by means of a perfectly straight “knife-edgef ground to a prism, of hard steel or agate, which is firmly connected with the beam, so that the edge coincides with the intended axis of rotation. In the best instruments the bearing plane is continuous, and the edge rests on it along its entire length; in less expensive instruments the bearing consists of two separate parts, of which the one supports the front end, the other the hind end of the edge. Every complete balance is provided with an “arrestment,” one of the objects of which is, as the name indicates, to enable one to arrest the beam, and, if desired, to bring it back to its normal position; but the most important function of it is to secure to every point of the central edge a perfectly fixed position on its bearing. So far all modern precision balances agree; but the way in which the point-pivots A and B of our fictitious machine are sought to be realised varies very much in different instruments. In Robinson’s, and in the best modern balances, the beam is provided at its two extremities with two knife-edges similar to the central one (except that they are turned upwards), which, in intention at least, are parallel to, and in the same plane as, the central edge; on each knife-edge rests a plane agate or steel bearing, with which is firmly connected a bent wire or stirrup, provided at its lower end with a circular hook, the plane of which stands perpendicular to the corresponding knife-edge; and from this hook the pan is suspended by means of a second hook crossing the first, matters being arranged so that, supposing both end-bearings to be in their proper places and to lie horizontally, the working points A' and B' of the two hook-and-eye arrangements are vertically below the intended point-pivots A and B on the edges. In this construction it is an important function of the arrestment to assign to each of the two terminal bearings a perfectly constant position on its knife-edge. How this is done a glance at figs. 3 and 4 (of which the former is taken from an excellent instrument constructed by L. Oertling of London, and the latter from an equally good balance, represented in fig. 5, made by Messrs Becker <fc Co., of New York) shows better than any verbal explanation. But what cannot be seen from these sketches is that the range of the arrestment is regulated, and its catching contrivances are placed, so that when the arrestment is at its highest place, the central edge is just barely lifted from its bearing, and the terminal bearings are similarly lifted from their respective knife-edges, so that the beam is now at rest in its normal position. In other balances, as, for instance, in the justly celebrated instruments of Mr Staudinger of Giessen, Robinson’s plane terminal bearings are replaced by roof-shaped ones (fig. 6), so that their form alone suffices to secure to them a fixed position on their knife-edges. Another construction (which offers the great advantage of being easy of execution and facilitating the adjustment of the instrument) is to give to the terminal edges the form of circular rings, the planes of which stand parallel to the central edge, and from which the pans are suspended directly by sharp hooks, so that the points A' and B' coincide with A and B respectively. In either case the terminal bearings are independent of the arrestment, which must consequently be provided with some extra arrangement, by means of which the beam, when the central edge is lifted from its support, is steadied and held fast in its normal position. In second and third class instruments even the central edge is made independent of the arrestment, by letting it work in a semi-cylindrical or, what is better, a roof-shaped bearing, which, by its form, assigns to it (in intention at least) a definite position.
In order now to develop a complete theory of the precision balance, let us first imagine an instrument, which, for distinctness, we will assume to be constructed on Robinson’s model, the knife-edges and bearings, &c., being exactly and absolutely what they are meant to be, except that the terminal edges, while still parallel to the axis of rotation, are slightly shifted out of their proper places. ' Supposing such a balance were charged with Ÿ = p' 0 + p' from the left, and P" = p" 0 + p" from the right knife-edge,—and it is clear that in this case also the charges may be assumed to be concentrated,—P' in a certain fixed point A on the left, and P" in a certain fixed point B on the right edge, and, consequently, the statical condition of the balance is the same as if the weights W, P', P" were all concentrated in one fixed point C o (fig. 7), the position of which, in regard to the beam, is independent of the extent to which the latter may have turned, and independent of the direction of gravity. It is also easily seen that in a given beam the position of C o will depend only on P z and P", and supposing P' to remain constant it will change its position whenever P" changes its value. The point C o will in general lie outside of the axis of rotation, and consequently there will in general be only two positions of the beam in which it can remain at rest, namely, first, that position in which C o lies vertically above, and, secondly, that position in which it lies vertically below the axis of rotation. Only one of these two positions can possibly lie within the angle of free play which the beam has at its disposal. The second of the two positions, if it is within this angle, can easily be found experimentally, because it is the position of stable equilibrium, which the beam, when left to itself in any but the first position, will always by itself tend to assume. The first position, viz., that of unstable equilibrium, is practically beyond the reach of experimental determination. Hence the points A, B, and S must be situated so that, at least whenever P7 z = P z 7 zz exactly or very nearly, the beam has a definite position of stable equilibrium, and that this position is within the angle of free play. To formulate these conditions mathematically, assume a system of rectangular co-ordinates, X, Y, Z, to be connected with the beam, so that the axis of the Z coincides with the central edge and the origin with the projection 0 of the centre of gravity on that edge, while the Y-axis passes through the centre of gravity. Let the values of the co-ordinates of the points A, B, S, C o (imagined to be situated as indicated by the figure)be as follows:— [table]
(The z's are evidently of no practical consequence.) To find x 0 and y 0 we need only again apply the reasoning which helped us in the case of the similar problem regarding the ideal instrument. Assuming, then, first, gravity,to act parallel to Y, we have (P' + P" + W) x η = P'7" - P7 z . Assuming, secondly, gravity to act parallel to X, we have (P' + P zz + W)y 0 = P7t z + P"Λ zz + Ws, .∙. for the distance of the common centre of gravity C o of the system from the axis of rotation, r≈ ,Jx 0 i + y 0 2 , and for the angle a through which the needle, supposing it to start from the zero-point, must turn to reach its position of stable equilibrium—
tan α- ι∕o^^ws o +P7ι z +P zz r · · · · < 3 ∕∙
If, in particular cases, one or more of the points A, B, S should lie above the X-axis, we need only consider the respective ordinates as being in themselves negative, and the equations (as can easily be shown) remain in force. Taking equation 3, together with what was said before, we at once see that if a balance is to be at all available for what it has been made for, and supposing two of the co-ordinates h', h!' to have been chosen at random, the third must be chosen so that, at least whenever P z exactly or nearly counterpoises P zz , Ws 0 + P z Λ z + V'h!' > 0. For if it were = 0, then, in case of P7 z = P'7, zz the balance would have no definite position of equilibrium, and if it were negative, y a would be negative, and the position of stable equilibrium would lie outside the angle of free play. Obviously, the best thing the maker can do is so to adjust the balance that h' = Λ" = 0 and l' = I", because then the customary method of weighing (see above) assumes its greatest simplicity, and, especially, the factor with which the deviation of the needle has to be multiplied to convert it into the corresponding excess of weight present on the respective pan assumes its highest degree of relative constancy. We speak of a degree of constancy because this factor can never be absolutely constant, for the simple reason that no beam is absolutely inflexible, and consequently h , as well as h!' is a function of P z , and P zz of the form h = h 0 + γP, where γ has a very obvious meaning. What is actually done in the adjusting of the best instruments is so to place the terminal edges that, for a certain medium value of P z + P zz , h ' + h" «= 0, so that the sensibility of the balance is about the same when the pans are empty as when they are charged with the largest weights they are intended to carry. The condition l ’ = I" also cannot be fulfilled absolutely in practice, but mechanicians now-a-days have no difficulty in reducing the difference y - 1 to less than ÷ 60 ⅞ c - σ , and even a greater value would create no serious inconvenience. We shall therefore how assume our balance to be exactly equal-armed; and, substituting for h' + h!' the symbol 2Λ, and understanding it to be that (small) value which corresponds to the charge, substitute for equation 3 the simpler expression tan α = W¾ + 2PΛ · · · · < 4 )’
which, on the understanding that P zz = P z + Δ, and that A is a very small weight, gives the tangent-value corresponding to P and Δ. Sometimes it is convenient to look upon the pans (weighing p 0 each) as forming part and parcel of the beam; the equation then assumes the form—
tonα- wv + ⅛* · · · (5) ’
where p = P - p u .
In a precision balance the sensibility, i.e., the tangentvalue of the deviation produced by Δ = 1, which is tan u α >> = I . /6),
Δ ws' + 2ph k 7 must have a pretty considerable value, and at the same time ought to be as nearly as possible independent of the charge. Hence what the equation (4) indicates with reference to a balance to be constructed is, that, so far as these two qualities are concerned, we may choose the weight of the beam as we like; and in regard to the sensibility which the instrument is meant to have when charged to a certain extent, we have even the free choice of the arm-length, because, whatever I or W be, if only the centre of gravity of the empty beam is brought to the proper distance from the central edge, we can give to the sensibility any value we please. What is actually done is so to construct the beam that its centre of gravity lies decidedly lower than one would ever care to have it, and then to connect with the beam a small movable weight (called the “bob 1 ’) in such a manner that it can be shifted up and down along a wire, the axis of which coincides with the Y-axis, and thus the value s 0 of the distance of the centre of gravity of the beam from the central edge be caused to assume any value, from a certain maximum down to nothing, and even a little beyond nothing. As to the relative independence of the sensibility of the charge, equation 5 shows that a given balance will possess this quality in the higher a degree the less the distance h of the central edge is from the plane of the two terminal ones, and, supposing h to be constant (i,e., the adjustment to be finished), the less the initial sensibility α rt exhibited by the empty instrument. Passing from one balance to the other, but supposing h and a 0 to remain constant, we readily see that the sensibility is the more nearly independent of the charge p in the pans, the greater the arm-length I is. From what has been said above, it would appear that by means of a balance provided with a gravity-bob, we could attain any degree of precision we liked, but evidently this is not possible practically, because in the actual instrument neither the knife-edges and their bearings nor the arrestment are what we have hitherto supposed them to be; and, consequently, both l' and Γ as well as h, instead of being constants, are variable quantities. Obviously, the nonconstancy of the ratio l': I" is the most important point, and to this point we shall therefore confine our attention. Let us imagine that the imaginary balance hitherto considered has been charged equally on both sides (with P = p 0 + p), so that its normal position is its position of rest, and then assume, first, that the middle edge (which hitherto has been an absolutely rigid line) is now a narrow and slightly, but irregularly, curved rough surface. The effect will be, that, supposing the balance to be repeatedly arrested and made to vibrate, the axis of rotation, instead of being constant, will shift irregularly between X = + λ and x = - λ where λ means a small length. But this comes to the same as if the central pivot were absolutely perfect, but had the common centre of gravity C o , instead of being fixed at x = 0, oscillating between x = ± λ 0 , . In other words, the balance may possibly come to rest at any position within a certain angle ± β, which, as an angle of deviation, corresponds to the overweight
⅞={2(P.+1>) + W}½.
Assume now, secondly, that, say, the right terminal edge was slightly turned so as no longer to be parallel to the middle edge. This in itself would not matter much, because although it might produce a change in the length of the right arm, this change would be permanent, and the arm-length again be constant, provided the hook-and-eye arrangement for the suspension of the pan, and the arrestment, were ideally perfect. But, practically, they are not, and, moreover, the knife-edge and its bearing are not what theory supposes them to be; and the effect, is the same as if the virtual point of application A of the charge Po + Pi instead of being at the constant distance I from the centre, oscillated irregularly between I + λ' and I - λ', where λ ’ has a similar meaning to that of λ 0 . The joint effect of the imperfections of the three pivots is that the indications of the balance, instead of being constant, are variable within ± e, where e means a small weight determined approximately by the equation—
<=∣{‰+2>) + W]λ,+ 2(i > 0 +p)λ} . (7X-
Hence, in a balance to be constructed for a given purpose, I must be made long enough to make sure of its compensating the effects of the λ ’ s, which, for a given set of knife-edges, and a given degree of absolute exactitude in their adjustment, may be assumed to have constant values. Evidently in a given balance e has nothing to do with the sensibility, and consequently it would be useless to increase the sensibility beyond what is required to make the angle β, corresponding to e (i.e., that angle within which the balance is, so to speak, in indifferent equilibrium), conveniently visible. To go further would, in general, be a mistake, because the greater the sensibility the more markedly it varies with the charge, the less is the maximum overweight which can be determined by the method of vibration, and, last not least, the more slowly the balance will vibrate, because the time of vibration t is governed by the equation —
<= ∕ ⅞ W+2(p 0 +p)~ _J„_
V Ws 0 +2λ,(p 0 +p) ‘ √¾ l where k is a constant which depends on the shape of the beam, and for the ordinary perforated rhombus is about = ⅜ , while R o stands for the length of the pendulum beating seconds at the place. Introducing the sensibility— a = — -, we have t = c Ja, where c is a constant.
Ws 0 +2Λ(p 0 +p) , n
4. Compound Lever Balances.— Of these numerous inventions—in all of which a high degree of practical convenience is obtained at the expense of precision—we must content ourselves with noticing two which, on account of their extensive use, cannot be passed over. We here allude, in the first place, to that particular kind of equal-armed lever balances, in which the pane are situated above the beam, and which are known as “Boberval’s balances;” and secondly, to those peculiar complex steel-yards which are used for the weighing of heavy loads by means of comparatively small weights.
In Roberval’s balance (fig. 8), the beam consists of a parallelogram, in which each of the four corners A, B, A', B' is a joint, and which by means of two joints situated in the centres of the two longer sides AB and A'B' is suspended from a vertical rod so that the two shorter sides A A' and BB' under all circumstances stand vertical. With these two sides the pans are rigidly connected; and the main feature in the machine is, that wherever the charge in the pan may lie, i.e., whatever maybe the virtual point of application of the whole charge P in regard to the vertical side of the beam, its statical effect is the same as if P was concentrated in a point D in the axis of the rod AA ’ or BB'. That this really is so is easily proved. Imagine the particle weighing P units to be rigidly connected with, say, A A,' but situated to the left of that line, and, whatever may be its distance from AA', when the beam descends through a certain angle, the vertical projection of the path described by the point D, i.e., its fall h, has the same value whatever its distance from AA ’. Hence the work done, say, against an elastic string tending to hold the beam in its place, invariably is = P7⅛ as it would be if D was situated in AA'.
The ordinary Decimal Balance is a combination of levers illustrated by fig. 9. a, c, b, d, e, g, h, f, are all joints or pivots; a and h rest on the fixed framework of the machine, and consequently indirectly on the ground; c rests on the lever ab. In the actual machine cd supports the “bridge,” which accommodates the load, while at f is suspended a pan for the weight. The pan is so adjusted that it balances the bridge. Suppose the load P to be placed so that its centre of gravity is at i, and a portion P c of P will press on the knife-edge at c, the rest P, l will pull at d and, with the same force, at g. Now, P e =
P. -~, equivalent to = P t pulling at b or e, equivalent to cd _ αb
P. = . ∙=r . — pulling at g. The dimensions are so chosen αά_ cd_ gh r 6 y that = = , hence the effect of P e at g is equivalent to ab eh l·d a weight P. — . The other portion of P, viz., P d , pulls cd — at d, and consequently also at g, with a force P. — . cd
Hence the effect of the total load is equivalent to P. = P units suspended at g, and if, for instance, gh — hf, one pound in the pan will counterpoise ⅛en pounds at any point of the bridge.
5. Torsion Balances. — Of the several instruments bearing this name, the majority are no balances at all, but machines for measuring horizontal forces (electric, magnetic, <kc.), by the extent to which they are able to distort an elastic wire vertically suspended and fixed at its upper end. In the torsion balances proper the wire is stretched out horizontally, and supports a beam so fixed to it that the wire passes through its centre of gravity. Hence the elasticity of the wire here plays the same part as the weight of the beam does in the common balance. An instrument of this sort was invented by Ritchie for the measurement of very small weights, and for this purpose it may offer certain advantages; but, clearly, if it were everto be used for measuring larger weights, the beam would have to be supported by knife-edges and bearings, and in regard to such application therefore (he., as a means for serious gravimetric work), it has no raison d’etre. See Electricity and Magnetism.
6. For Hydrostatic weighing machines see the article Hydrometer. (w. d.)
