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IX.— On the 1 -partitions of X. 



By the Rev. Thomas P. Kirkman, A.M., Rector of Croft- 

 with- South worth, and Honorary Member of the Literary 

 and Philosophical Society of Manchester. 



[Read April 1th t 1857.] 



The subject of partitions of numbers has been greatly ad- 

 vanced within the last two years. Euler was, as I believe, 

 the first to show that the number of ways in which x can be 

 made up of a, b, c . . . m, each element being repeated an 

 indefinite number of times, is the co-efficient of r* in 



I 



(l_;-)(l_r*)...(i-r») ; 

 and thus the problem was reduced to the decomposition of 

 rational fractions. This view of the matter, however, led to 

 no practical results, until, by the profound researches of Mr. 

 Cayley, printed in the current volume of the Philosophical 

 Transactions, and by the more decisive discoveries of Pro- 

 fessor Sylvester, who has, if I mistake not, thoroughly worked 

 out the idea of Euler, and given the results of it in a calcu- 

 lable form, the problem of partitions has been directly solved, 

 without any aid from induction. That is, Professor Sylvester 

 has given the general formula both for the non-circulating and 

 the circulating part of the expression of the k-partitions of X, 

 in the Quarterly Mathematical Journal for July, 1855. 



The direct solution is obtained by an analysis of no ordi- 

 nary difficulty ; and I think it not unlikely that, for practical 

 use, the inductive method given by mc in the twelfth volume 

 T 



