MAJOR P. A. MACMAHON ON THE COMPOSITIONS <>K NTMUKK'S. 

 6othat 



therefore 1-X^ = 1 _ XHl +XH f -X'H a + .. 



I ~~ A 



and thence 1 _ 1 H! XH 3 + X 2 H 3 ... 



^_ j_ j 



-XH 1 + X 2 H 2 -X 3 H 3 +. 

 Now let functions A A A A 



-"1 > -"-2 > -"-3 > -"-4 i 



be connected with u tr TT 



ti l , 1 2 , tl 3 , 11 4 , ... 



in the same way that 



!, a 2 , a 3 , a 4 , ... 



are connected with , , , , 



ll %, 3, 4, 



so that A TT 



A! = M! , 



A 2 = Hi a H 2 , 

 A 3 =H 1 3 -2H 1 H 2 +H 3 , 



then I __ 



1 \rt I \2tT \3TT 



1 Atli + A. rl 2 A H 3 + ... 



and = l + (H 1 -XH 2 +X 2 H 3 -...)(l+XA 1 



1 ~~ J\. 



On the dexter the co-factor of X' is 



which has the value 



Since ^-h> 



is a well-known identity in the elementary theory of symmetric functions. Hence 



-L- = 1 + A, + XA,+X S A 3 +X 3 A 4 +...; 

 1 A 



or, as we may write it, 



1 



, + XA 2 -|-X 2 A 3 +X 3 A 4 +.. M 



+ X : '(H 1 4 -3lVILfH/+2H,H 3 -H 4 )+... 



