58 Prof. Cay ley on the Distribution of Electricity 



Comparing this with 



W +A +1 - jr^i (Sl)> + '- 1 X^ + « + 



(X B+1 -\)(-&c + £)}, 

 where the term in -j J- is 



= (^ +2 -l)[^"^)] + 



(X«+'„X)[-<c 2 ~a 2 -^)+(c 2 -a 2 )( C -^)] ? 

 it is to be observed that the quotient of the two terms in \ j- 

 is in fact a constant ; this is most easily verified as follows. 

 Dividing the first of them by the second we have a quotient 

 which when ^ = c is 



(X" +1 -l)(-^) + (X w -X)(-m 2 ) _ a 8(x"+'-l+x w -\) 



B+1 - 



-X)(-c(c 2 



-a 2 -6 2 )) 

 a 2 (\-f 

 (c 2 -a 2 - 



1) . 



-6 2 )v 5 



+, -\)(c>- 



-a 2 - 



6 2 ) 



and when 



^=0 is 













(X w + 1 -l)c(c 2 - 



• a 2 -6 2 ) 



(X n+1 - 



-l)(c 2 -a 2 



-^ 2 ) 





(\" +> - 



■l)6 2 c + (\ n 



c 2 -a 2 - 



-6 2 



-l + \ n+1 - 



-X> 2 





these two values are equal by virtue of the equation which de- 

 fines X ; and hence the quotient of the two linear functions 

 having: equal values for m = c and # = 0, has always the same 



. c 2 -a 2 -b 2 . 



value ; say it is = , 2 , . -i \ ~ • Hence, observing that a + d = 



u + 8j = c 2 — a 2 — b 2 , the quotient, c n a 2 -\-d n (c — x) divided by 

 X+l c 2 -a 2 -& 2 _1 



or we have the required equation 



c n a 2 + d n (c-%) = p . (« B+1 a> + £ w+1 ). 



Considering now the functional equations, suppose for the 

 moment that g is =0 ; the two equations may be satisfied by 

 assuming 



