1886.] e\ log e (l±v), (l+x) m . 421 



which can be re-written in one way only in the form 

 ly 1 + 2y. 2 +... + t(y t + l) + ...+r % =r > 



7i+ %+•••+ (7« + 1 ) + —+ %= 7 '- s > 



showing that the term corresponding to the solution 



& = 7i> &=7 2 > — &=y t ,>~ (3 r = y r , 



can be derived by breaking up that term in 

 v l« l + 2a,+ ...+rg r where |l ai + 2a 2 +...+ra r = r, 

 ajaj ...a r !l a '2 a * ...r^' l a i + a 2 +...+ « P = r— 5, 



for which a l = y 1 , a 2 = y 2 , ... a t = y t +l, ... a r = y r ; and from that 

 term only. 



Hence the coefficient of w r " s_1 in — - is 



n 



i 



ft! £,!...#! 1*2*. ..»*■' 



for all possible solutions of the pairs of equations 



/3 1 + /3 2 +...+/3 r =r-s-l 

 and 1&+2&+ ..-. +r/3 r =r.-!; 



or /8, + /3 2 + . . . + £ r = r - s - 1 



and 1£ + 2/3 2 + . . . + r@ r = r - 2 ; 



or @ 1 + 2 +...+p r = r-s-l 



and l^ i+ 2/8 2 +.'..+r/8 r = r-i; 



or j3 1 + /3 a +...+fi r = r — 8-l 



and 1/^ + 2/3,+ ... + r/3 r = r-.s - 1. 



Now consider the coefficient of n r ~*~ l in 

 c + c t + c 2 + ... +c r _!. 

 It consists of parts contained in 



c r _,_ 1 + c r _, + c r _ ri . 1 + ... -Hc^. 

 Therefore 



/remembering that the coefficient of w" in c q is \ 



S 1 where ^ + 2*, + .-• + ?«,= <7 ), 



\ a^. a 2 \ ... a q ll a >2 a * ...q a v { a t + a, + ...+ « a =p/ 



