8fi MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 



We proceed in this manner until finally we combine 01 consecutive pairs and 

 throw the remainder of the units together, thus producing 



N numbers, each containing n 0+1 parts. 



Hence the compositions that present themselves are included in those enumerated by 



N,, N_,, , N B _ 9+1 . 

 We have to consider the product 



in all of its permutations and for every system of values of 



V> ' S 2> > S 9> 



such that 



Sj + .Sjj +...+.<(, = n. 



Hence, from considerations of symmetry, and attending to the modits operandi of 

 the multiplication theorem, we find that the whole of the compositions enumerated by 



N N N 



"I ^n-l) > -^n-S-M 



present themselves. 



Hence the linear function we seek is a linear function of 



N>J "NT "NT 



n-fl + l) L ~n-6+2> > " II -L"*) 



and it remains to determine the coefficients. 



The number of products, including permutations, 



N(1'-)N (!")... N (IV), 







which we have to consider, is ecpual to the numbers of compositions of n into 6 parts, 

 viz., it is 



(n-e) > 



each of these produces 



S0-V 



m 



N numbers, each containing n m parts. 

 There are thus 



\n-6l \ m 



N numbers, each containing nm parts. 



But there are only 



{n 1 



m 



