MAJOR P. A. MAC.MAHON ON THE COMPOSITIONS OF NUMBERS. 127 



and thence 



(N m -N._ -+I )- s = (N.-N.-O."'- 



For n = 6, these relations can be verified by the tables for all values of m, 

 Art. 107. Similarly the theorems derived from 



can be at once written down. 



| It is worth noting that this operator relation can, by putting D, = /;A,, l>e written 



W...D.D.,... 



we can write down the corresponding relation between the numbers N, 

 It will be found that /xr 



is a linear function of vr vrp) 



**m. l"i 1* n,l 



Hnd (N.-N...* 



a linear function of N'Vi-', N^ ,.-, 



and that the same obtains when instead of 



V 



we take ,.-1. 



v l- x j 1 



Art. 108. From the operator relation 







where tt,,, i, -,, . . . are numerical coefficients that may be determined. 



Thence is derived the relation u , ,, 



^".ih r* 



giving a hint to put N .,- = N.-' 1 symbolically, 



and then /xr > \ w , /M i \i 



- - ' 8ym bolically. 



Art, 109. It must now be remarked that, since 



