MAJOR P. A. MA.MAHON ON THE COMPOSITIONS OF M-.MUKKS. 



131 



and, generally, the product of any number of operators is expressible in the required 

 linear form. 



Art. 115. With the object of connecting this theory of the numbers N M with that 

 of the n motors N (><l><\..), the generating function 



will now to expanded in ascending powers of X, the coefficients of X toing functions 

 of the homogeneous product sums 



The point of departure is the elementary formula 



1 Oi + 

 Remarking that 



l 



I write 



equivalent to writing 



and 



then 



1 Oi + Oj Oj-r ... 



n a +... = (1 a,)(l 



/, xx r 

 (l-X)a. for a., 



/. . x, e 

 (1 X)'. for a., 



(l-X)'A. for h.-, 



and, as tofore, write 

 so that 



whence, solving for A, 

 and 



= u suppose ; 



(1 X)a a +(l X)-a 3 ... = A; 

 . 



u = 



A= 



i _i 

 1-X u ' 



where - r- is the generating function under consideration. 



1 A. 



Wnte H t =h l+ h,+h 3 +h t +... 



H, = h 3 



H 4 = A 4 



a 2 



