172 174] STEADY FLOW IN CONDUCTORS. 343 



from which, putting F 4 = 0, and using the last three equations, 



C sl C 3 , 

 <? <? 



A 

 From this we obtain the current in 34, 



r* ri /ri ri ri ri \ 



The current vanishes and the conductors 12, 34 are conjugate, 

 if 



(7 31 (7 42 = 41 32 , that is ~ = -^ . 



This arrangement is used for measuring the resistance of a 

 conductor in terms of three known resistances. A battery is 

 inserted in one of the conductors 12, 34, and a galvanometer in the 

 other, which is called the bridge wire. The resistances in the 

 other branches being varied until the galvanometer shows no 

 current, the condition of conjugacy is attained. In practice we 

 wish to know how much current will pass when the condition of 

 conjugacy is deviated from by a certain amount, in order to 

 determine the accuracy with which a resistance can be measured. 

 We therefore have to calculate the determinant A. 



174. Resistance of Linear Conductor of Variable. 

 Section. If the cross-section of a conductor is infinitesimal, and 

 equal to &>, we may write for the total current 



dV 



-ii' 



dti = Xw , , 

 dn ds 



s being the length of the conductor measured from a certain point. 

 Integrating with respect to s from Sj to s 2 , 



v f 2 Ids , [ 2 ds 

 YI F 2 = =// , 

 J i Xw J t Xo> 



and the resistance is given by 



R = 



= r ds 



} ! Xft) ' 



