THEORY OF THE PARTITIONS OF NUMBERS. 159 



in a form showing its mode of derivation from 



(l-xf 



We now see that, when n' = p+ 1, we get a form which may be written 



and then, when n' = p + 2, the form is 



where the numerator of the function last written is 



Hence 



ip+l\ 

 = r 2 }u pl -u p3 , 



= (p +l 



U 



P+1,P~1 ( t,_]J llpltlpp , 



\p 

 I 



These relations are satisfied by 



2p 

 v 2-1 J\l 



so that the result of the summation is 



> ( 



pq ~p\ <z-i / \p-qj' 



n 







\fn\f 2n\ +1 1/n+lW 2n \ +2 , y lf2n-2\f2n\ ^ 



'- 1 -'* f n\ 2 yU-sj* ( >n(n-i)(or 



and there is no difficulty in showing that this in fact is equal to 



-l\ 



