THEORY OF THE PARTITIONS OF NUMBERS. 351 



Since the sum of the coefficients in IL ( <x ; a, 6, c) is 



(a-6+ 1) (a-c + 2) (b-c + 1) 

 it WAS at first conjectured that the expression might be 



hut this neither satisfies the functional equation nor verifies in simple particular 

 cases. 



If, in the formula 



we put IL ( QO ; a, b, c) equal to the expression above and then put r = 0, we obtain 

 the known result for GF( oo ; , />), as may be readily seen by putting the expression 

 in the form 



We are therefore justified in putting IL( oo ; a, b, c) equal to the expression with 

 an added term which contains the factor (c). 

 Write, therefore, 





By working out several particular cases I was led to the conjecture 



F ( : > M = jj) {*>-(a-b)-.r-'(b-c)} ; 

 and I then found that the expression 



does, as a fact, satisfy the functional equation. 



Art. 9. Having thus, beyond doubt, established the forms of IL ( oo ; a, b) and 

 IL ( oo ; a, b, c), I proceed to a study of the functional equations. 



In the equation 



= (a+l)IL(oo;a-i,6)4-(b)IL(cx 5 ;,6-l)-(a-fl)(b)IL(oo;a-l > 6-l), 

 put 



X-{IL(cx>;a,6)-(a-fl)IL(oo;a-l,6)}=V 1 (oo;a,t); 



