MECHANICS. 



15 



it will assume' the position represented 

 by the faint line, the arm S'c being; as 

 much above the horizontal line uc as it 

 was when on the other side above the 

 horizontal line a'e : but that is evidently 

 as much as the arm S c was below the 

 horizontal line a c. Hence, it is quite 

 apparent, that if the positions of the 

 points S and S' below and above the 

 line a c before and after reversion be 

 noted, half the angle S' c S will be the 

 deviation of the line S c or S S' from 

 the perpendicular aa'. 



(37.) A process somewhat similar to 

 this serves to determine the deviation of 

 the index from the direction of the line 

 g m (Jig. 24.) Suppose that the previous 

 adjustment has been made, and that the 

 line S S' is perpendicular to g m ; but 



Fig. 24. 



still that the index m v deviates from 

 the direction of g m by the angle b m v. 

 It is necessary to determine practically 

 whether any such deviation exists, and 

 if so, to what amount. 



As before, suppose the beam sus- 

 pended against a vertical plane, and the 

 position of the point v marked. Let the 

 beam be reversed, and the index will 

 assume the position mv', deviating as 

 much to the left side of the true direc- 

 tion mb as it before deviated to the 

 right, The point v 1 being then marked, 

 the angle v' m v will be twice the devi- 

 ation of the index from its true position. 



The common commercial balance is 

 usually sustained upon a loop of metal 

 on which the beam rests by a knife 

 edge. In this case, when the beam is 

 unloaded, the index ought to settle ex- 

 actly between the sides of the loop, and 

 it should always be in this position 

 when equal weights are suspended from 

 S and S'. 



(38.) The several adjustments which 

 we have now described being made, it 

 will be evident that, when equal weights 

 are suspended from S S', the beam will 

 maintain its horizontal position. For 

 the perpendicular distances of the ver- 

 tical lines through S and S', which are 

 the directions in which the weights act, 

 are equal, being, in fact, S c and S c'. 

 Hence these distances, when multiplied 



by the equal weights, will give equal 

 products, and therefore the weights 

 will have equal tendencies to turn the 

 machine in opposite directions round the 

 centre of motion m, and, consequently, 

 they will mutually destroy each other's 

 effects, and the instrument will maintain 

 the position it had when unloaded. 



But let us consider what would be 

 the consequence if unequal weights 

 were suspended from S and S'. Let AY be 

 suspended from S, and W from S', and 

 let W be the greater. Let the common 

 length of the arms S c and S' c be a. 



The tendency of the weight W to de- 

 press the arm S c is measured by the 

 product of this weight, and the length a of 

 the arm S c orW x a : and the tendency 

 of the weight W to resist this is the 

 product of ~the weight W' and the arm 

 S'c, or W x a. Now, as W is greater 

 than W', the product of W and a must 

 be greater than the product of \V' and 

 a, and therefore the tendency of W to 

 depress the arm S c is greater than the 

 tendency of W' to resist that, and there- 

 fore the arm S c will fall and S' c will 

 rise. 



It appears, therefore, that if the ba- 

 lance be properly constructed, and the 

 several adjustments which we have 

 pointed out be attended to, it will only 

 maintain that position in which the 

 beam is horizontal, and the index ver- 

 tical, when loaded with equal weights ; 

 but that if either weight be greater than 

 the other, it will always incline in the 

 direction of the greater weight. 



The sensibility of a balance is mea- 

 sured by the smallness of the difference 

 of weights which turns the index from 

 its unloaded position m o, and by the 

 greatness of the deviation v m o from 

 the unloaded position which that dif- 

 ference produces. To explain this more 

 fully, let us suppose that, W being greater 

 than W, the beam rests in equilibrium 

 in the position represented in fig. 25. 



. 25 



