82 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 



Art. 26. Considering next p different numbers, defined by the partition 



we have, by a previous definition, 



where a 1} a 2 , o 3) ... are each <1 and such that 



S =p. 

 I have written 



NP> 



instead of N M , in order to specify the number of objects (or numbers) subjected to 

 permutation. 



Art. 27. I shall now prove that 



where in 



the number of objects subjected to permutation is n, and the summation is in respect 

 of all permutations such that the sum of the first m numbers in the descending 

 specification is equal to p. 

 For, by Art. 16, 



2N (!.,...,...) = "' N (a^... a w l) ; 

 hence 



and, by the multiplication theorem, 



( j p+l)N(a 1 cr a ...a m )N(l) = N(a l a f ...ftl) + N(a,a,.. . 

 so that 



i 

 and since 



SN 







whence, by summation, 







but since 



