198 PROCEEDINGS OF THE AMERICAN ACADEMY. 



bers of (8), let e^ — Gr be substituted for its equal Ck hk. It will tlien 

 be found that all the terms which contain r disappear, since the co- 

 efficients of Cr and of 0^ r^ vanish identically, and that 



^^^....(^.^ 00) 



iV„ 4 = 1 b^ b^ 



where the si2,n of summation introduces once only every value of 

 {eic — Cj)'^, in which k is ditfereut from^, and neither k nor / is greater 

 than n. 



Since w does not involve r, it {w) is the work done inside the 

 battery when its poles are not connected by any external resistance. 

 Equation 10 shows that w cannot be negative, and that it is different 

 from zero unless all the cells in the battery have equal electromotive 

 forces. The expression for w may be written in a form due to 

 Slouginoff, 



* = "'"" '/, (H) 



where Cq and h^ are the electromotive force and the internal resistance 

 respectively of the equivalent cell. 



It is known that if the poles of a battery formed of n cells be con- 

 nected by an outside resistance r, the current in the external circuit 

 will be tlie same, whether the cells be joined up in series or in multiple 

 arc, provided that 



iV„ 2 (e,) - Q^ 



(12) 



or, if the cells are all alike, provided that r =z b. It is worthy of 

 notice, however, that the efficiencies of the battery are different in 

 the two cases. If the cells are joined up in series, the efficiency of 

 the battery is 



r 



but if they are joined u)) in multiple arc the efficiency is 



C- r 



so that 



F, C^r+C'^ib,) 



(15) 



