MAJOR P. A. MxcMAHON ON THE COMPOSITIONS OF NUMBERS. 121 



Art. 89. Every symmetric function identity has corresponding to it a relation 

 between the operators ; thus corresponding to the set 



(I) 4 = (4) + 4(3l) + 6(2*) + I2(21 a ) + 24(l*), 

 we have the set 



= 2 (l') D +(2) D , 

 ,) = (</,) + 3 (rfAJ+rf, = 6 (l) D + 3 (21) D +(3) D , 



= (24) (!)+ 12 (21 i ) D + 6 (2% + 4 (31) D + (4) D , 

 and so on. 



Art. 90. Also, corresponding to the set 

 2a, = ,*-, 

 6a s = Sj* 3*,.'. l 



Ac., we have the set 



2D, = (</,') = (<W-d, t 



24D 4 = (d, 4 ) = (d,) 4 -< 

 and so on. 



Art. 91. For the special operand 



1 



1-A 



these operator relations assume a special simple form which is of great importance in 

 the theory of the generating function. 



For , . / \._ii._i /, j A \ / \.-ii.i-i 



rf,A = ( )' 1 6' (1 bA.) = ()' l b' 



or, qud the above operand, , _ (_y-it ) '-i f i 



and thence, from a set of relations given above, 



Art. 92. By means of these we can now arrive at a most important series rf 

 relations. 



VOL. ocvn. A. B 



