78 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS, 



and not at all upon the numbers 



Observe that w l and the final number of the composition may or may not be unity, 

 and that every composition of n may be written in the form 



If 



3=S=S'= =0 



mtwwww ) = -- nl 



2!' 



and, in particular, 



wherein iv 2 , ;r 3 , ... w m _, are non-unitary, but w it w m may or may not be unitary. 

 As a simple example take 



u\ = 1, s, = 4, w 2 = 1, 

 so that 



a verification. 



Art. 21. A more general theorem is yielded by 



n! 



wherein 



are given integers and the summation indicated on the dexter is in respect of the 

 whole of the compositions of the numbers 



WL 'a, >3, w 4 , 

 where 



03=?<?, 1. w a 2, w 3 2 and u\ l. 



The value of the sum is thus 





