THEORY OF THE PARTITIONS OF NUMBERS. 355 



iiii,' with 4 , m times successively, upon P, I obtain 



and I draw the inference that 



(c-l)(c)(b+l) and (c-l)(c)(a+2) 

 are solutions. 



Further, operating with O t , m times successively, upon P,, I obtain 



and it thence appears that 



(a+l)(a+2)(c) and (b)(b+l)(c) 

 are solutions. 



Again, operating with O t , m times successively, upon P,, I obtain 



and the conclusion is that 



(a+l)(a+2)(b+l) and (b)(b+l)(a+2) 



are solutions. 



No other fundamental solutions are obtainable by operating with O a , O 6> and O c 

 upon P,, P a , and P 3 , and clearly we have no need to consider the other operators 

 because of the relations between them. 



We have thus six fundamental solutions 



Art. 13. The known solution of the functional equation from which these solutions 

 have been derived can now be expressed in terms of these. Since it has been found 



that 



O." (l) a (2) IL ( oo ; a, 6, r) = P l -*-+'P,+*-'- +3 P Sl 



we have 



(l) a (2) IL ( oo ; a, 6, c) = P.-*?, 



and, putting m = in results obtained above, it appears that 



P, = (a+l)(a+2)(b+l)-*(b)(b+l)(a+2), 

 P, = (a+1) (a+2) (c) -*> (b) (b+1) (c), 



2 z 2 



