489 



superintendence, of the transformed equation for the quintic equa- 

 tion. The several results are given in the present memoir ; and for 

 greater completeness, I reproduce the demonstration which I have 

 given in the former of the above-mentioned two notes, of the general 

 property, that the function of y is an invariant. At the end of the 

 memoir I consider the problem of the reduction of the general 

 quintic equation to Mr. Jerrard's form ,r'-f ax+b = Q. 



December 12, 1861. 



Major-General SABINE, R.A., President, in the Chair. 



In accordance with the announcement made from the Chair at 

 the last Meeting, the question of Mr. Sievier's readmission was put 

 to the vote, and was decided in the affirmative. The President 

 accordingly declared that Mr. Sievier was readmitted into the Society. 



The following communications were read : 



I. "On a Series for calculating the Ratio of the Circumference 

 of a Circle to its Diameter." By AMOS CLARKSON, Esq. 

 Communicated by Professor STOKES, Sec. U.S. Received 

 September 27, 1861. 



The ratio (?r) of the circumference to the diameter of a circle may 

 be calculated by the following series : 



= 8f. __ 1_ 2 2.4 1 



31 3.10 3.5.10 2 3.5.7.10 3 "j 



7 3.10 2 3.5.10 1 3.5.7.10 8 



This series may be thus established. We have, as is well known, 



and denoting by c the arc of which the tangent is /, 



Put 



