PEIRCE. — BATTERIES IN MULTIPLE ARC. 



195 



Let C be the current which flows through the outside resistance r, 

 and let C^. be the current which flows, in the same cychc direction 

 as C, through the /;th cell. Then, if \ — X^ r, 



k = I 



p = n 



p = i 



(1) 



(2) 



Let the determinant of the coefficients of the C's in the set of n 

 linear equations of which (2) is an example, be denoted by A„ ; then 



A„ = /•« 



1 + Ai 1 1 1 



1 1 + Aa 1 1 



1 1 1 + A, 1 



1 1 1 1 + A4 



1 + K 



or 



A = r" S 



It is evident that 8„ ^ , would be equal to the product of X„ + ^ and 

 8„ plus the product of all the X's from Xj 10 X„. 



Let 3f„ represent the product of all the b's. 



Let N„ represent the sum of the n products of the 5's taken n — 1 

 at a time. 



Let P„ J represent the sum of the n — 1 products formed of all 

 the b's except b^ taken ?^ — 2 at a time. 



Let Q,^ represent the sum of the n products formed by multiplying 

 every e by all the ^'s except its own. 



Let S„^^. represent the sum of the products obtained by multiplying 

 every electromotive force except that of the kih cell by all the b's 

 except its own and b^. 



Then A^ = M;, + r iV„. (3) 



The determinant formed from the determinant of the coefficients 

 (A„) of the C's in the n linear equations of which (2) is a type, by sub- 

 stituting the corresponding e's for the elements of the ^th column shall 

 be called A„ j. 



