Page:Encyclopædia Britannica, Ninth Edition, v. 3.djvu/278

262  Fig. 2. Diagram illustrating the theory of the Lever Balance. Theory of the Lever Balance (fig. 2).—In developing the &quot; theory &quot; 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 arc 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 on the axis of rotation stands perpen dicular on that plane. Suppose now two weights, P and P&quot;, 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&quot; respectively. But two such points are equivalent, statically, to one point (weighing P + F&quot;) situated somewhere in D within the right line connect ing A with B. Suppose the beam to be arrested in its &quot; normal position &quot; (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 1 : 1&quot; to be given, on the ratio F : P&quot;. If P I &#61; P&quot; 1&quot;, 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 I is constant, I&quot; can be made to vary and its ratio to I 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&quot; 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 then read off -^, and have P &#61; P&quot;. But, practically, the i i body to be weighed cannot be directly suspended from A, but must be placed in a pan suspended from A, and consequently the weighty 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 &#61; P p. 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 I (or some convenient sub- multiple or multiple of I ) in length, and so to adjust p and number the scale, that when the sliding weight P&quot; 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 .../&amp;gt;, 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&quot; units to use a set of standards weighing F, $ P&quot;, -^^ P&quot;, 77jV?7 P&quot;, &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, con structed 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 -g-jjVtfth f it s va l ue -

Second Method.—We so construct our instrument that both I and I&quot; have constant values, and are nearly or exactly equal to each other, and provide it with pans, whose weights p and p &quot; are so adjusted against each other that p I p&quot;l&quot;, 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 com bining the several pieces with one another, enables us to build up any integral multiple of the smallest difference of weight 8 we care to determine, a set, for instance, which virtually contains any term of the series O OOl, 002, 0.003 lOO OOO 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 combina tion of standards p&quot; which, when placed in the right pan, establishes equilibrium to within o. Evidently—

P &#61; ^. . . . . (i). In the case of purely relative weighings, there is nothing to I&quot; ( I&quot; hinder us from adopting units ( e.g., -y grammes ) as our I  l J unit of mass, and simply to identify the relative value of p with the number p&quot;. But even if we want to know the absolute value of p in true grammes, we need not know I&quot; the numerical value of . All we have to do is, after having determined the value of p in terms of &#61;-, to reverse v the positions of object and standards, and, in a similar manner, to ascertain the value p&quot; which now counterpoises the unknown weight p lying in the right pan. Obviously I&quot; i&amp;gt; . p &#61; p&quot; j &#61;p&quot; j, whence (//) 2 &#61; p&quot;p&quot;, and p &#61; *Jp*p&quot;, for which expression, if the two arms are very nearly of equal length, we may safely substitute p &#61; 1(p&quot; +/&amp;gt;/ ) Or, instead of at once finding the counterpoise for p in stan dards, we may first counterpoise it by means of shot or other 