120 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBEES. 



and writing 1 A = b, it is written 



1 c^ + fcoa-frV, + ...+(-)* b*- l a* ' 

 I 



1-A' 

 where 



A = a 1 _ 

 For the present purpose we may consider k to be infinite, and write 



A = ! bn 2 + 6 2 a 3 ... . 

 Art. 87. Taking the symmetric function operators 



, = ai1aj203 ... = , 



and an auxiliary fictitious equation 



r being an infinite number, it is necessary to remind the reader of the relations 

 existing between the operators. 



Successive linear operations of , 7 , 



dto a,, ,, ... 



are denoted by placing them in separate brackets, thus, 



but when they are multiplied, as in TAYLOR'S theorem, so as to produce a single 

 operator of higher order, they will be placed in one bracket, thus, 



Art. 88. Let monomial symmetric functions of the fictitious relation 



af-D^-' + DjZ'-'-... = 

 be denoted by a partition in brackets with subscript D, thus, 



( )D- 

 Then I have shown, in a previous paper, 



and, in general, 



d, = U, 2 -2D 2 = (2) D , 

 d, = . . . = (s) D , 



