420 Mr Hill, On the series for [May 10, 



this becomes 



1 



which is of similar form to the terms in the coefficients of the 

 powers of n in the expression for c r , but here 



l/3 1 +... + (t-l)/3 t _ 1 + t/3 t +(t + l)/3 t+1 +...+r/3 r 

 = la x + ... + (t - 1) a M +*(v 1) + (« + 1) a m + .... + ra r = r- 1, 

 and /3 1 +...+A. 1 +A + /3 m +...+/3 r 



= of 1 + ... + or i _ 1 + (a t - 1) +or (+i +...+ a r = r-s-l. 

 In these equations t must be > 1, and also since none of the 

 /3's are negative and one at least must be positive 



A + ,.. + A-+...+i8 r <lft+'.-...+.^,+ ...+r/9 rl 

 .'. r — s — 1 <r — t, 



.'. t<S + l. 



Hence 1 < t < s + 1. 



Hence on breaking up all the terms included in 



v "lg 1 + 2«,+ ...+rg r where (U 1 + 2a 2 +...+ra r = r r 

 a t \ a 2 \ ... ol t \ l«i2 a «...r a ' ' l «i+ « 2 + •■• + a r = r-s, 



there is obtained the sum 



1 



25 



&I &!...&! 1*2*...**' 



where U& + 2& + ... + r& = r-*, 



1 <«<* + -l. 



And in this last summation terms corresponding to all possible 

 solutions of the equations 



1& + 2&+. .. + r/3 r = r-t, 



& + /3 2 + ...+ r = r-8-l, 



1 <t<s + l, 



are included. 



For take the solution t = y v /3 2 = y 2 , . . . ft = y t , . . . ft = y r , 

 then 



l7x + %+ ••■ + ty t + ... + ry r = r - £, 

 7i+ 7« +.•■■ + 7«+ •••+ y r = r-s-l, 



