THEORY OF THE PARTITIONS OF NUMBERS. 93 



when = 61, for all values of a. Then 



/.M- (l)(2)...(a+b-l) ^(a 



. 



(1) 

 .r*(a- 



..(b-l)' (1) 



4- ... 



(1) 



The right-hand side has a 6+1 terms; assume that the sum of the last 2> terms 



may he written 



, . 

 ' 



an assumption which is obviously justified when p = 1 ; then the sum of the last 

 p+l terms (p > a b) is 



...- 



(1) (2) ... (b+p) . (1) (2) ..: (b) (1) (2) ... (b+p-1) . (1) (2) ... (b-1) ' 



and this on simplification proves to be 



which is a justification of the assumption. Hence the right-hand side of the expression 

 of L ( oo ; ab) is, leaving out the first term, 



_. 



...a....(b)^ 

 leading to 



L(oo ab) (l)(2)...(a+b-l) _ ^(a 



--l)' (1) 



_ 

 -(2)(3)...(a+l).(l)(2)...(b)' (IT 



This result, being true when b = 0, is thus established universally. The outer 

 function is of the required form, and the inner function is 



