104 MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS. 



The known value of h n is thus given by a law identical with the multiplication law 

 of this paper, and the expression of A B in terms of 



is completely given by 



, 



fl n Cj. 



This new statement, of a well-known law, immediately suggests the generalization 

 to which I proceed. 



Observe that n and 1" 



are zig-zag conjugate compositions. 

 From the relation 



is now deduced 



and, since , ; 



and we again observe that 



pq and \ p I 



are zig-zag conjugate compositions. 



Hence writing (1^21^) = (pq)', 



a (py> = "(PI)' ' 

 and, in general, I have established (but reserve the proof for another occasion) that 



where / \ , v 



(Pip-i---), (PiP*-.-) 



are zig-zag conjugate compositions. 



Art. 61. The theorem has an interest of its own, but it is also of vital importance 

 in this investigation. This importance consists partly in the circumstance that the 



functions 



'W... 



are those which naturally arise in the present theory of permutations. The present 

 theorem enables the immediate expression of them in terms of the elementary 



symmetric functions 



i, 2 , a 3) ... 



and thus they may be more easily dealt with by symmetric functions differential 

 operators. In fact, the homogeneous product sums 



can be made to disappear from the investigation ; but, as will be seen, it is sometimes 

 advantageous to retain them wholly or in part. 



