88 MAJOR P. A. MAoMAHON ON THE COMPOSITIONS OF NUMBERS. 



As equalities may occur between the numbers p, q, r, ..., I take, for greater 

 generality, the specifying composition 



(PW-)- 



It will be seen later that the order of occurrence of the parts of this composition is 

 immaterial, so that we may consider the parts p lt p a , ... to be in descending order of 

 magnitude and the specification to be denoted by a partition 



tew-)- 



E.g., we obtain the same results for each of the six assemblages, 



aa^yyy, a/3/3/3yy, ctfiftyyy, 

 the specification of each assemblage being 



(321). 

 Every permutation has a descending specification. 



has the descending specification 



In the case considered in Part I. the assemblage of numbers had the specification 



(1-) 



since there were no similarities, and the numbers N(...) were expressed in terms of 

 the coefficients obtained by the multinomial expansion 



(a 1 + o 3 +a 3 +...) n . 

 E.g., we found 



N (a) = coefficient of symmetric function (a) in the expansion, 



where, in the first case, = n, and in the second, a + b = n. 



In a usual notation let 



i, fh, fh, 



denote the homogeneous product sums, of the successive orders, of the roots of the 

 equation ---' --'-+ .... = ; 



we may say that, in Part L, the auxiliary generating function was 



(a 1 + a 2 + a 3 +...)' 1 = V, 



a u s, a a, being the roots of the equation. 



Art. 33. In the present case the auxiliary generating function is 



f) *>}> *>/) * 

 ... n p, "P, "A 



as will appear. 



