THEORY OF THE PARTITIONS OF NUMBERS. 103 



This proof rests upon the assumption that the law can l>e shown to hold for 

 GF (nsl, 2, x) for all values of s from to n 1, and for all values of /i. Now 

 suppose the law to hold for GF (I, 2, v) for all values of v inferior to n ; then it 

 obviously holds for GF (nsl, 2, .<t) for all values of s inferior to n, and thence, as 

 has just been proved, it holds for GF (/, 2, n) ; but the law does hold when v = or 1, 

 and thence by induction the law holds in general. This method of proof seems to he 

 of application only when m = 2, for then only can the function L, ( <x, 2, n) be 

 identified with a form GF (l' t 2, n') where the sum of l'+n' is less than n. 



Art. 26. I turn again to the relation 



in order to establish relations between the functions GF (I, m, n) and the sub-lattice 

 functions L,(oo,m, n). The relation, as it stands, exhibits GF (I, m, n) as a linear 

 function of the sub-lattice functions, but giving I the special values 0,1,2,... in 

 succession, we obtain 



GF (0, m, n) = L ( <, m, n) = 1, 



I -*- j f(fj * " / 1 j V I - * I \ " ) " fc > /) 



f 2 , ,^ - (5EAL( nm + 2 ) + ( mn + 1 ) 



(2 ' W ' n) - (1)(2) + " (1) 



(mn+l)...(mn+/Lt) (mn+1) ... (mn+^-1) T 



GF (p., m, n) = - +- - LII+ +L M 



(!)...(/A) (!)...(/* 1) 



and thence 



L =l, 



,- ) 

 L, = GF (2, ,, n)- (^5+llGF (1 , m, n) + * <4J1 , 



-l,m^ 





