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



We can now determine the highest symmetric function, in dictionary order of the 

 parts, which occurs in the development of h alx . This, by the known theory of 

 symmetric functions, is obtained from the form 



a (ate...)' 



by expressing (abc...)' as a partition and taking the Ferrers conjugate (abc...)" ; 

 then we see that no symmetric function, prior in dictionary order to 



(abc...)", 

 can appear. 



Also the highest integer in (abc...)' 



is the lower limit of the number of parts, occurring in the partition of a symmetric 

 function, arising from the development of 



E.g., since 



, 



"141 ~ 



we arrange 21 2 2 as a partition, obtaining 2 2 1 2 , and taking the Ferrers conjugate from 

 the graph 



we reach (42) as the highest symmetric function in dictionary order that occurs in 

 the development of h in . 



Hence N(l4l) e = N(14l) M = 0. 



(See the table of weight 6. ) 

 Numerous relations such as 



^i4i + ^5i = h iv +h a 



can be verified by the same table. 



Art. 64. Before proceeding to establish the multiplication theorem, the generali- 

 zation of that in Part I., it is necessary to examine the mode of operation of the 



differential operator p. 



*/ 



upon a product , . Wl 



n pi n p, > 



vv 



It is clear that T. 7 ; 



1>A = h p - a . 



In the paper it was shown that 



where a' a" denotes a composition of a into two parts, zero not excluded, and the 

 summation is for every such composition. 



