290 THE ELECTRIC TELEGRAPH. 



in proportion to the resistances of the elements, we can set 

 RJ = R 2 = R 3 = 0, and the expressions become, 



E, 



Dividing the two last equations by the first, 



E, 

 whence 



E 2 : E, = r 2 - r, : r, 

 and 



E 3 : E, = r 3 r, : r, 



proportions which imply that the number of elements E x in 

 the circuit of smallest resistance being given, the numbers 

 E 2 , E 3 , &c., to be added to each of the other circuits must bear 

 the same proportion to E L which the difference between their 

 resistances bears to the smallest resistance. More generally 

 if the resistances of the lines were equal, the single battery 

 would produce a like current in each of them ; but if not, 

 the battery power which must be added to the lines of 

 superior resistances to produce the same current as in the 

 smallest line, must be exactly proportioned to the superiority 

 of their resistances. 



41. WTieatstone' s Balance. On the same laws depend the 

 mathematical proof of the truth of the beautiful and useful 

 system of resistance measurement invented by Professor 

 Wheatstone, a description of which appeared in the Philo- 

 sophical Transactions of 1843. 



The poles of a battery, E, Fig. 137, are connected to the 

 points of union c and d of parallel circuits, r lt r 3 , and r 2 , r 4 , 

 and between some points, a and b, in these two conductors a 

 wire, r 5 , is inserted. The current takes the course indicated 

 by the arrows, and several complete circuits are formed, for 



