14 



MECHANICS. 



standards of weight. This is one of the 

 most interesting and generally useful 

 applications of the lever, and assumes 

 various forms, according to the nature 

 and magnitude of the substances whose 

 weights are to be determined, and to 

 the degree of accuracy which is required 

 in the result. 



The most usual form of the balance 

 is that of a lever of the first kind with 

 equal arms ; so that the substance to be 

 weighed being suspended from one arm, 

 and the weights assumed as standards 

 of comparison being suspended from 

 the other, the equilibrium will neces- 

 sarily be established when these weights 

 are equal (14). 



This, however, is only the general 

 principle of the common balance. In its 

 construction there are various circum- 

 stances to be attended to which are of 

 considerable importance. 



(33.) The lever which forms the ba- 

 lance, and which is called the beam, 

 should be so constructed that its centre of 

 gravity should be immediately under the 

 axis or centre of motion. For if the 

 centre of gravity were itself the centre 

 of motion, the beam would rest indif- 

 ferently in any position ; whereas the 

 equality of the weights is determined 

 by its assuming the horizontal posi- 

 tion. If the centre of gravity were 

 above the centre of motion, the least dis- 

 turbance would cause the beam to 

 upset. (Treatise I. chap, iv.) 



The centre of gravity being, by the 

 construction of the beam, beneath the 

 centre of motion, the line joining it with 

 the centre of motion will, when the 

 beam is unloaded, always settle itself so 

 as to be in a vertical direction. 



(34.) The substance to be weighed, 

 and the weights with which it is com- 

 pared, are placed in dishes suspended 

 from points at the extremities of the 

 beam, called the points of suspension. 

 These points should be so placed, that 

 a straight line drawn joining them shall 

 be perpendicular to the straight line 

 which is drawn joining the centres of 



Fig. 22. 



gravity and motion, and so that it shall 

 be divided by that line into two equal 

 parts. 



That is, if S and S' (fig. 22.) be the 

 points of suspension, m the centre of 

 motion, and g- the centre of gravity of the 

 beam ; the lines S S' and m g should 

 intersect at c at right angles, and S c 

 should be equal to S' c. 



The beam being thus constructed, 

 the pointy will, by the properties of the 

 centre of gravity explained in Treatise I. 

 settle itself vertically below m, so that 

 m g shall be perpendicular to an hori- 

 zontal plane. The line S S', being per- 

 pendicular to g m, will in that case be 

 horizontal. 



(35.) In order to exhibit, in using the 

 balance, the direction of the line m g 

 (fig. 23.) a needle or index is attached to 

 the beam, which sometimes plays upon a 

 graduated arch ; and when it is directed 

 to that point of the arch which is in a 

 vertical line passing through the centre 

 of motion, the line S S'will be hori- 

 zontal, and the line mg, which is the 

 direction of the index, will be vertical. 

 This, as we shall presently see, is the 

 position of the balance which indicates 

 the equality of the weights suspended 

 from S and S'. 



(36.) The practical determination of 

 all these circumstances in a beam is not 

 difficult. Under any circumstances, the 

 line mg, when the beam is at rest, will 

 be vertical. In order to determine 

 whether S S' is in that case horizontal 

 or perpendicular to mg, let the beam 

 be suspended against a vertical plane, 

 and mark the points on the plane at 

 which S andS' are placed. Then lift the 

 beam off its centre and reverse it. If it 

 be found that S' exactly takes the place 

 of S, and S of S', then the line S S' is 

 horizontal, and at right angles to m g, 

 but otherwise not. To explain this more 

 clearly, let us suppose, that the line S S' 

 is, in the first instance, not perpendicular 

 to mg, but that it deviates from the per- 

 pendicular ac by the angle S ca. Let 

 the position of the points S and S' on 

 the vertical plane against which the 

 beam is suspended be marked, and let 

 the beam be reversed. When reversed, 



