62 MEASUREMENT OF COUPLES. 



rectilinear; the middle O of the cross-piece AA' rests at O 1? and the 

 wires take the positions BAj and B'A' r If z is the distance 

 'Oj = BD, we have 



ib cos 



(') ,o 



= l~ (b a) 2 Afdb sin - - . 

 2 



From which we have 



(2) zdz = -ab sin d6. 



Let Q be the moment of the horizontal couple necessary to keep 

 the system in its new position. The work of this couple, for an 

 infinitely small rotation, is equal and opposite to the work of gravity 

 on the weight O of the system ; we have then 



?ab sin OdO 



z 



or, taking into account equations (j) and (2), 



ab sin 



. 

 (3) Q = - sm0 = 



/ I (b - a} 1 Wb 



V 1 ~~^ /*" 



/ / (b-aY tab . 2 6> 



sin * - 

 2 



If the condition a = b is fulfilled, that is to say, if the two wires 

 are parallel in the position of equilibrium, we get 



(4) 



The ratio - and - are usually very small; in this case the 



denominator only differs from unity by an infinitely small quantity 

 of the second degree, and we may write 



(5) 

 putting 



