MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBKKR. 133 



Art. 116. The preceding pages show thut the coefficient of 



A 



in the expansion of 



-X) '< a -(l - 

 is equal to 



the summation heing for every partition 



(pqr...):' 

 and this, from the theory of the numhers 



is equal to 



SS...S/ 



PI PI P- 



the summation being for all integer values of 



or, the same thing, for the compositions of all numhers into exactly m parts. 



Hence 



SAfc-A,-H t< 



= A 3 =H I11 =H 1 s -2H,H a +H s , 



a remarkable result. 

 Art. 117. Since 



we find, putting X = 1, 

 and, since 



