THEORY OF THE PARTITIONS OF NUMBERS. 



357 



In general, put 



and consider the product A.B^C y D 4 where A, B, C, D are in fixed alphabetical order 

 and a, ft, y, 8 is some permutation of 3, 2, 1, 0. 

 We find 



and the effect of the symbols q a , q 3 , q t is to multiply by (b J3+2), (c }/+!), 

 (d 0} respectively. Hence 



since 



= 1 + 2 + 3. 



It is thus established that each of the products in question is a solution of the 

 functional equation. 



Art. 16. Hence the determinant, which is a linear function of these products, viz. : 



(a+l)(a+2)(a+3), (b) (b+1) (b+2), (c-1) (c) (c+1), s(d-S)(d-l)(d) 

 *(a+2)(a+3) , (b+1) (b+2) , (c)(c+l) , x(d-l)(d) 



X , XT , X , 1 



and I shall show that this determinant, divided by the determinant 



1, 1, x, x 3 



X, I, 1, X 

 X 3 , X, 1, 1 

 X 9 , X 3 , X, 1 



is the actual expression of IL ( oo ; a, b, c, d). 



Art. 17. I first take the test of the sum of the coefficients which we know otherwise 

 to be 



(a-b + l)(a-c+2)(a-d+3)(b-c+l)(b-d+2)(c-d+l). 



