MAJOR P. A. M.v. MAHON ON THE COMPOSITIONS OF NUMBERS. 87 



different N numbers, each containing nm parts, because 



m 



is equal to the number of compositions of n into nm parts. 

 Hence, each N number, comprised in 



N.-., ' 

 will occur 



w ;^ n -m -r )tirae8- 



it v 



("') 



Hence the required linear function is 



v/ n ~ m ~ 1\ M 



or 



and the final result is 



X n! 



where 



Sw, = n, Si> = ^. 



PART II. SECTION 3. 



Art. 32. In the preceding pages we have had under view the permutations ot n 

 different numbers. As I am now taking in hand the general case of numbers which 

 possess any number of similarities, I find it convenient to slightly alter the point 

 of view. 



Let , Ay, -. 



denote numbers in descending order of magnitude, and suppose there are 



p number equal to a, 



so that, placed in descending order, the assemblages may be written 



I say that the assemblage is sj>ecined by the composition 



(pqr...). 



