PEIRCE, — BATTERIES IN MULTIPLE ARC. 197 



Ci. 



Q 



h I N 



which defines Stepanoff's equivalent cell * already mentioned. 'Ihe 

 difference of potential between the poles of the battery is 



V= "^ (7) 



When r is made to increase indefinitely, F approaches as a limit the 

 electromotive force Q I N^oi the equivalent cell, and (7^ approaches 



— — 2i^r= — , the current in the ^'th cell when the poles of the battery 



are not connected by any external conductor. 



If there is a battery of electromotive force E in the external circuit, 

 r, the quantity JiJ must, according to the direction of this external elec- 

 tromotive force, be added to or subtracted from the second member of ' 

 each of the equations of which (2) is an example. If a battery of n 

 cells joined up in multiple arc be itself connected up parallel with a 

 cell of electromotive force £J and of internal resistance B, E -^ B must 

 be added to the numerator of the fraction which forms the second mem- 

 ber of (6), and r -i- B to the denominator. Upon an examination of 

 these cases, it appears that, if a battery of cells joined up in multiple 

 arc be itself connected up parallel or in series with another battery, 

 or if it be used for compensation purposes, it will exert the same in- 

 fluence upon the currents and diff'erences of potential in parts of the 

 circuit external to itself as its equivalent cell would exert. 



The internal work done m the battery when its poles are connected 

 by the external resistance, r, is 



W,Ji\c,H,). (8) 



k = 1 



The internal work done in the equivalent cell would be 





2(Q 



(9) 



Let v) be the difference between W^ and W(, and in the equation 

 found by subtracting the members of (9) from the corresponding mem- 



* See also SlouginofE, Carl's Repertorium, XVI. 539. 



