MAJOR P. A Mv MAHON ON THE COMPOSITIONS OF NUMBERS. 85 



Further, we obtain a result of this nature for all values of 



such that Sw, = n,2v = 0; and, by addition, we obtain 



ST .? ! ; K- r ^ r = linear function of numlien N, 



where Sw. = n, 2> = 6. 



We have now to determine the linear function of numbers N which appears on the 

 dexter. 



If one such number be j^ / ^ \ 



it is evident that / \ 



is some composition of the number n. 

 Consider the product of 6 factors 



N(1")N(1*)...N (I 1 *), 

 where S* = n. 



The process of multiplication produces N numbers of 6 different kinds. 

 In the first place we throw all the units together, 



N (l" + VK" + '*) ( 



one N number containing n parts. 



In the second place we combine a consecutive pair of factors and throw the 

 remainder of the units together, thus producing 6-1 N numbers each containing 



n 1 parts, viz., 



N(l"- 1 21 



N i*' + *~ 1 



In the third place we combine two consecutive pairs (including, of course, a 

 consecutive three) of factors and throw the remainder of the units together, thus 

 producing , e _ 



( 2 



N numbers each containing ?i-2 parts, viz., the series of which one is 



N (l''-' 

 Notice that, if *, = 1, this becomes 



