THEORY OF THE PARTITIONS OF NUMBERS. 173 



we find 



2 + a 3 C 3 + . . . 4- a n C H . 

 To prove this relation I will show that any factor 



1 -- _(><) of C 

 a t +s t^ 



is also a factor of 



2 + . . . + a n C a . 



First observe that 1 -- - is a factor of CL unless m is equal to s or t 



. a t + st 



Therefore consider merely 



a t C t +a,C t . 



The factors of C which involve either a t or a s or both a, and a t are 



,. \m = t-\ I n \m=s-lf -. ,, = n / 



i -- = ] n (i -- ^ ) n (i -- ^ n i 



,=i \ a m + s TO/ ,=+i \ a m + sm/ P =, 



q=-l / 



n i n i n i 



Hence, disregarding a common factor, C t involves the factors 



a t a,+s t l'"^ l a m a, + s m m= ^' l a m a s + s m p =" a s a p s+p 

 a t + st l =i a m +sm m =t+i a m + sm /)= +i a s s+p 



1=1 di + tl ?=s +i a t + q t 1 ?=<+! 

 and C, involves the factors 



a t a, + s t+l m= ^~ l a m a s +s m+l " l= ^ l a m a s + s m+l ^" a, a p +p s 

 t m =i a m + sm m =t+i a m + sm P = s +i a t +psl 



i=\ ai + tl q= s +i a t t + q q=t+i a t t + q 



Discarding common factors from these expressions we find that C t involves the 

 factors 



. _ 



a t +s t 1 a,s+n 



*n 

 x ( 



a t +s-t a t 



