148 Royal Society : — 



fore seem to bear a resemblance in its physiological relations to 

 vegetable, rather than to animal glucose. 



The following communications were in part read : — 

 " Researches on the Partition of Numbers." By Arthur Cayley, 

 Esq., F.R.S. 



The author discusses the following problem : — "To find in how 

 many ways a number q can be made up of the elements a, b,c . ., 

 each element being repcatable an indefinite number of times." The 

 solution depends upon a pecuhar decomposition of an algebraical 



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



of factors, the same or difierent of the form 1 —a-'", and upon the 

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

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

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

 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. 



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

 Cayley, Esq., F.R.S. 



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- 



(!) V if) 3C 



thod reduced to the form coeiF. x'" in ^, where ^ is an algebraical 



fx fx 



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



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



of factors of the form \—x^, 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 Quantics." 



May 10. — The Lord Wrottesley, President, in the Chair. 



The following communications were read : — 



"An Experimental Inquiry undertaken with the view of ascer- 

 taining whether any force is evolved during Muscular Contraction 

 analogous to the force evolved in the Fish, Gymnotus, and Torpedo." 

 By Henry Foster Baxter, Esq.* 



" On a simple Geometrical construction, giving a very approximate 

 Quadrature of the Circle." By C. M. Willich, Esq. 



Let AB be a quadrant of a circle A, B, C. In the arc BC place a 

 chord BD equal to the radius, so that the arc BD is one of 60°. 



* This communication will apjiear in our next Number. 



