377 



fraction^-, where the denominator fx is the product of any number 



of factors, the same or different of the form 1 x m t and upon the 

 expansion by means thereof of the fraction in ascending powers of x. 

 The coefficient of the general term is expressed in terms of circu- 

 lating functions, such that the sums of certain groups of the coeffi- 

 cients are severally equal to zero ; these functions the author calls 

 prime circulators. The investigations show the general form of the 

 analytical expression for the number of partitions, and they also in- 

 dicate how the values of the coefficients of the prime circulators 

 entering into such expression are to be determined. 



II. " Further Researches on the Partition of Numbers." By 

 ARTHUR CAYLEY, Esq., F.R.S. Received April 14, 1855. 

 With Postscript. Received April 20, 1855. 



The memoir contains a discussion of the problem " to find in how 

 many ways a number q can be made up as a sum of m terms with 

 the elements 0, 1, 2,. ... k, each element being repeatable an inde- 

 finite number of times." The number q may without loss of gene- 

 rality be taken to be equal to (km a), and the expression for the 

 number of partitions of this number (km a) is by a peculiar me- 

 thod reduced to the form coeff. x m in -^-, where % is an algebraical 



/* f* 



fraction, the form of which depends on the value of k, but which does 

 in anywise involve the number m ; the denominator fx is the product 

 of factors of the form 1 x e , and up to certain limiting values of a 

 the fraction is a proper fraction. The author remarks in conclusion 

 that the researches were made for the sake of their application to 

 the theory developed in his " Second Memoir upon Quanrics." 



