522 



Major J. Herschel. 



this point it were balanced by ^0, any increase of torsion caused by 

 increase of 0, however small, would be answered by no corresponding 

 augmentation of the opposing force. The upper system would obey, 

 6 would increase, but the resistance would diminish instead of increas- 

 ing. In short, 90° would represent a position of unstable equilibrium. 



This state of things is prevented by the torsion of the parallel wires, 

 which act in aid of the weight and defer the condition of unstable 

 equilibrium to a point some degrees beyond 90°. The perception of 

 this may or may not have been present to the designer ; but it may 

 obviously be made use of advantageously as follows : — Let 2p%6 denote 

 the torsion of the two parallel wires when turned through an angle 0. 



Then the forces which balance are (P +p)— . sin f 2p 2 0, and ^0 ; and 



R 



the utility of the instrument, depending in the first place on the abso- 

 lute constancy of p Y 4>, depends also on the possibility of exactly equali- 

 sing to this the other force by varying either or P+_p. It is in this 

 respect exactly analogous to a balance, the desired relation of to 

 being equivalent to the demand for horizontality in the latter. The 

 sensibility of such an equipoise will be measured by the smallness of 

 the change in P +p requisite to produce, or to correspond to, a given 

 small change of 0. 



Let (P+JP) 4 • sin0 + 2/o 2 0=F, 



and Bp=— cosec . e ((V -\-p)~ . cos O + Zp* A S0. 



r \ R / 



When 6=0°, hp is infinite ; i.e., no change of weight will affect 0. 



Similarly when 0=180°. But when = 90°, hp = 2 P o—hO. At this 



- r 



point no relation would subsist between hp and h0 but for /> 3 ; i.e., if the 

 equilibrium were independent of the torsion of the parallel wires. As 

 it is, the equilibrium is a compound one, and is disturbed by a change 

 of weight, the effect of which is inversely proportional to the torsion of the 

 parallel wires, in apparent contradiction to the first of the above pro- 

 positions regarding torsion. If the torsion of these wires be doubled, 

 the addition of the same small weight hp will cause a change of to 

 one-half the amount. The sensibility of the instrument is therefore 

 proportional (within limits) to the fineness of these wires ; bat this 

 must not be carried so far as to bring the position of unstable equili- 

 brium too near to 90°, or else the stability of the balance will become 

 too delicate and observation too difficult. 



