194 PROCEEDINGS OF THE AMERICAN ACADEMY. 



VII. 



ON THE PROPERTIES OF BATTERIES FORMED OF 

 CELLS JOINED UP IN MULTIPLE ARC. 



By B. O. Peirce. 



Presented May 9, 1894. 



If three galvanic cells the electromotive forces of which are ei, e^, 

 and 63 respectively, and the internal resistances, bi, b^, and 63, be joined 

 up parallel to each other, the battery thus formed is equivalent — so 

 far as its ability to send electricity through an external circuit is 

 concerned — to a single cell of electromotive force e,,, and of internal 

 resistance 6q, where 



eo = 



ei b^ bs + ^2 bi bs + €3 bi b^ , 61 b^ b 



b,= 



bo bs + bsbi + bi b^ 5, b^ + ^i ^3 + ^i b.^ 



A similar statement * may be made about a battery of any number 

 of cells connected in multiple arc. It is evident, however, that the 

 efficiency f of such a battery of unlike cells is less than that of a single 

 cell which would do the same amount of useful work under the same 

 external circumstances. 



I have had occasion of late to consider some relations between the 

 strengths of the currents which pass through the different members of 

 a given battery of unlike cells joined up in multiple arc. Many per- 

 sons must have used the equations which I found convenient in this 

 work, but I cannot find that any one % has taken the trouble to print 

 them all. I therefore give a few of them here, with some well 

 known formulas introduced for the sake of clearness. 



Let the internal resistances of n cells, which are joined up in mul- 

 tiple arc with each other and with a conductor of resistance r, be 

 ^n ^2> ^3 • • • respectively, and the electromotive forces be e,, €•>•, e„ . . . 



* Stepanoff, Journal Russ. Phys. Chetn. Soc, XII. 38. 

 t Slouginoff, Journal Russ. Phys. Chem. Soc, XIV. 2. 



X I have not had access to the papers of Messrs. Stepanoff and Slouginoff 

 quoted above. 



