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



Secondly, since 



...) (Mr ...) = D p D ? D r ... 



where, on the dexter, the operand is a function formed from the functions /i,, h 3 , h, 3 , . 



in the same manner as 



#0{() 



is formed from the operators 



Hence 

 where 



Art. 51. I now write 

 so that 



and it appears that 



Ni n i tr \ _ 

 I CfrUC . . . J (pqr.. .) 



is the true generating function of the numbers 



for the permutations of assemblages of numbers of all specifications. 



In fact, 



h^... = 2N (a&c...) (MP ...> . (pqr...} ; 



and the expansion of 



hate... 



as a linear function of monomial symmetric functions gives a complete account of 

 numbers 



Art. 52. Before proceeding to a rapid examination of this new and most important 

 symmetric function , 



" abc 1 



never before I believe introduced into algebraic analysis, I give complete tables of the 

 numbers N (...) as far as n = G. 



n = 2. 



= specification. 



