Page:Encyclopædia Britannica, Ninth Edition, v. 3.djvu/282

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 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 a,, exhibited by the empty instrument. Passing from one balance to the other, but supposing h and a a 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,itwould 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 arrest- inent are what we have hitherto supposed them to be ; and, consequently, both / and I" as well as h, instead of being constants, are variable quantities. Obviously, the non- constancy of the ratio I : I" is the most important point, and to this point we shall therefore confine our attention. Let us imagine that the imaginary balance hitherto con sidered has been charged equally on both sides (with P = p + p), so that its normal position is its position of rest, and then assume, first, that the middle edye (which hitherto has been an absolutely rigid line) is now a nar row 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 rota tion, instead of being constant, will shift irregularly between x = + A. and x where A, means a small length. But this comes to the same as if the central pivot were abso lutely perfect, but had the common centre of gravity C , in stead of being fixed at x 0, oscillating between x = A ! . Ill other words, the balance may possibly come to rest at any position within a certain angle /?, which, as an angle of deviation, corresponds to the overweight

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, be cause 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 arrest- nient, 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 P + p, instead of being at the constant distance I from the centre, oscillated irregularly between I + A. and I - A. , where A. has a similar meaning to that of X . 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 deter mined approximately by the equation—

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 X 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 /?, corresponding to e (i.e., that angle within which the balance is, so to speak, in indifferent equilibrium), con veniently 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—

where k is a constant which depends on the shape of the beam, and for the ordinary perforated rhombus is about = ^ , while R stands for the length of the pendulum beating seconds at the place. Introducing the sensibility— a = - TTT -- . , we have t = c Ja, where c is a constant.

4. Compound Lever Balances.—Of these numerous inven tions in all of which a high degree of practical conveni ence is obtained at the expense of precision we must con tent 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 pans are situated above the beam, and which are knowu as " Roberval s balances;" and secondly, to those peculiar complex steel-yards which are used for the weighing of heavy loads by means of compara tively small weights.

FIG. 8. Eoberval s Balance.

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 sus pended from a ver tical 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 ma chine 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, AA, 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 = P/i as it would be if D was situated in AA .