THEORY OF THE PARTITIONS OF NTMMKIIS. :;;:; 



Substituting the first of these for IL (I ; a, 1>, c) in the functional equation of 

 order 3 I find 



F _ 



but the solution of the equation 



F, = <jF,_! 



is clearly 



F, = <,<;_ !<,_j ... 



so that, in the present case, 



F -(1+1) (1+2) (0(1+1) (l- 

 -- a 



(i_ 2 )a 

 so that I obtain a fundamental solution 



Art. 25. Similarly I arrive at five other fundamental solutions, 



(a+1) (a+2) (c) (l)(l+l) (1+2) 



(b)(b+l)(a+2) (l)(l+l) (1+2) 



(c-1) (c) (b+1) (l- 



and I next seek, guided by previous work, to construct the function IL (I ; a, b, c) by 

 a linear function of these six solutions. 

 It is natural to write 



(a+1) (a+2) (b+1) (l+l) 2 (1+2) -x (b) (b+1) (a+2) (1) (1+1) (1+2) 



-x (a+1) (a+2) (c) (1) (l+l) (1+2) +x s (c- 1) (c) (a+2) (l) a (l+l) 



+x* (b) (b+1) (c) (l-l) (1) (1+2) -x* (c-1) (c) (b+1) (1-1) (I) (l+l), 



with a denominator 



(l) a (2) (l+a+1) (l+a+2) (l+b+1), 

 3 A 2 



