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



the coefficient of rf' 8 ' 1 



m c^is z, ^ ^^ . lai (j ,_ s _ 1 y r _ s _ 1 



. flflL + ... +(r-s-l)a r _ s _ 1 = r-s-l 



where i . , t 5 



1 1 



mc P ..is i^j a ^,. lai ( r _ 5 )a_ 



where |la i+ ... + (r-*K_ s =r- S 



1 1 



rt a x \ a r _ t \ l a i (r—t) a r-t 



. , f Iol + ...+ (r-t) a r _ t = r — t 



where \ * v ' • 



| a t + . .. + «,._, =r-s-l 



^ 1 1 



mc^is z^j a ^,. lttl ( r _!)-,_! 



where |1« 1+ ... + (r- 1) a r _ 1 = r- 1 



[ a t + . . . + a,._! = r - s - 1 



Now consider the equations 



which written more fully are 



l/3 1 + 2 y 8 2 +...+(r-^)/9,_ e + (r-i + l)^. m +...+r/3, = r-i 

 & + &+...+£._, + &_ m + .,.+ /3 r = r-s-l. 



In these it is impossible that 



Pr-t+l> Pr-t+2' ■•■ Pr 



can be different from zero, for if any one of them were, the left- 

 hand side of the first of these equations would be greater than the 

 right, because none of the /S's are negative. 



Hence these equations have the same solutions as the equations 



l/3 1 + 2/3 2 +..:+(r-t)l3 r _ t = r^t 



&+ & + ...+£._, =r-s-l 



(leaving out of account the zero values of /3 r _ m , fi r - t ^ ... fi r , which 

 in no way affect the terms following the symbol X in the previous 

 part of the investigation). 



