56 Prof. Cay ley on the Distribution of Electricity 



The determination of the functions <f>(x) and <§(x) gives thus 

 the complete solution of the question. 



I obtain Poisson's solution by a different process as fol- 

 lows : — Consider the two functions 



a 2 (c — x) a^ + b 



e»-y-w = S+a su PP° se ' 



and 



b 2 (c—x) KX + /3 



o - 2 — - , = -* suppose ; 



cr — <r — ex 7 7^ + ll 



and let the nth. functions be 



a^ + b n i * n x + /3 n 

 c n x + d„ 7 W « + 8 n 



respectively. 



Observing that the values of the coefficients are 



(a, b ) = ( -a 2 , a 2 c ), and (*, @) = (-b 2 , b 2 c ), 



|c, d| |-c ;C 2 -5 2 l | 7 , S| |-c,c 2 -a 2 | 



so that we have 



a + d = * + S, =c 2 -a 2 -b 2 , ad-bc = aS-£7 ? = a 2 b 2 , 

 and consequently that the two equations 



(X+l) 2 = (a + d) 2 (X+l) 2 = (a + S) 2 

 X ad— bc ? X 6tS—/3y 



are in fact one and the same equation 



(X+l) 2 _ Q 2 -q 2 -Z> 2 ) 2 

 X ~ a 2 b 2 



for the determination of X, then (by a theorem which I have 

 recently obtained) we have the following eqjations for the 

 coefficients 



(a*> b»), (xn, (3 n ) 



lc n , d n | \7n, Sn\ 



of the nth. functions; viz. these are:— 



a^ + b n =^^(|±^y _1 {(X^^-l)(a^ + b) + (X»-X)(-d^ + b)}, 



c^ + d M = „ „ {(\ w + 1 -l)(c^ + d) + (\«-X)( car-a)}; 



and similarly 



V + /3n=^^(^) n "\(X w + 1 -l)(^ + /3) + (X»-X)(-^ + /3)^ 



7n^ + 5 w = „ „ {(x»+i-l)( 7 - l . + S) + (X,»-\)( yx-a)}. 



