MAJOR P. A. MM MAHOX ON THE COMPOSITIONS OF NUMBERS. 77 



placed between a mt and &, ; we obtain 

 SN (a t a,. ..,.. .&!&,.. .&.,) 



n! 



n! 



By varying the order of the factors, other summations, leading to the same 

 numerical result, can be effected. 



Art. 20. Consider next the multiplication 



(x, + 2)!(*,+2)!(x 3 +2)! 

 x {N (I)}"-- 1 N ( l" +a ) {N (1)}*-*N (1" +J ) {N (I)} 1 "'-' N ( r< + ) {N (I)}"--' ; 



wherein, Si^ + S* = , 



Wi, w a , w 3) w t are numliers not less than unity, 

 *i. s a, "a are any numbers, zero not excluded. 



The result of the multiplication consists of numbers N (...), such that there is 



(i) A composition of u' t followed by x, units, succeeded by 

 (ii) A composition of w a followed by x a units, succeeded by 



(iii) A composition of w :t followed by .s 3 units, succeeded by 



(iv) A composition of w 4 ; 



and the dexter is the sum of all such nuinliers N (...). 

 Denoting this sum by 



SN (w^W WH0 4 ), 



we find that its value is 



n! 



since each number N(...) occurring in the product on the sinister has unity for its 

 value. 



Hence, in general, the remarkable theorem, 



showing that the sum depends merely upon the numbers 



