1892.] Theory of the Compositions of Numbers. 293 



and hence these two fractions, in regard to the terms in their expan- 

 sions which are products of powers of s iai , s 2 x z , . . . , *, must be 

 identical. This fact is proved by means of the identity 



1 1 - s l (2^ + o 2 + ... + a n ) | 1 1 s- 2 (2oi + 2o 2 + 



1-2 (S* iai - 

 multiplied by 



1 4. 5> Kl Kl . . . , < Kl l . . . , t 



a. T - , a " K i K 2 Kn t 



(1-fcvJ .................. (1-S K< ) 



where 



and the summation is in regard to every selection of t integers from 

 the series 



1, 2, 3, ... n, 



and t takes all values from 1 to n 1. 



This remarkable theorem leads to a crowd of results which are 

 interesting in the theory of numbers. One result in the pure theory 

 of permutations may be stated. 



Calling a contact a u ott a major contact when u > t, the number of 

 permutations of the letters in the product 



which possess exactly s major contacts is given by the coefficient of 

 in the product 



{a 1 + X(o 2 + ... +a n )}^{ 



and, moreover, is equal to the number of permutations for which 



r 2 + r 3 + ... +r w = s, 

 rt denoting the number of times that the letter * occurs in the first 



places of the permutation. 



Section 5 gives an extension of the idea of composition and of the 

 foregoing theorems. 



The geometrical method of " trees " finds here a place, and, lastly, 

 there is the fundamental algebraic identity 



