MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 73 



Art. 11. Again, for the summation 



where the composition begins with fit leant x units, we easily obtain the value 



n\ 



The Multiplication Theorem. 



Art. 12. A fundamental property of the numbers N(...) will be established from 

 elementary considerations ; it will, later on in the paper, be generalised. 



Let 



N(o 1 o > ...o k ) 



be derived from the permutations of p different integers, and 



N (..+.) 

 from the permutations of n p different integers ; it is to be shown that 





where on the right the reference is to the permutations of n different integers. 

 Out of the n numbers 



1, 2, 3,...n, 

 we can select 



( 

 numbers in 



ways, and arrange each selection, so as to have a descending specification 



(!,... a.), 

 in 



N(rr,a,...a.) ways; 



the remaining numbers can be arranged, to have a descending specification 



(, + l.+J. !+) 



in 



N (. +1 a, +1 ...a, + ,) ways ; 



placing the latter to the right of the former there appears 



arrangements. 

 VOL. ccvu. A. 



N K a -.) N (. 



