On forms of continued Fraction for Circle-quadrature. 331 



It cannot be surprising that the values of the reflection -con- 

 stants change with the distance of the opaque particles, since 

 the transparency and colour of thin layers of the same metal 

 also depend on it. (Faraday, Exp. Res. iv. 391.) 



The dependence of the optical properties of a quantity of more 

 or less transparent and non-transparent particles on the distance 

 of the latter seems to me hardly compatible with the assumption 

 that absorption and dispersion of light are determined solely by 

 a mutual vibration of the molecules of the body, occasioned by 

 the motion of the aether. 



Wiirzburg, Aug. 20, 18/3. 



XL. On Sylvester's and other forms of continued Fraction for 

 Circle-quadrature. By Thomas Muir, M.A., F.R.S.E., 

 Assistant Professor of Mathematics in Glasgow University*. 



IN the Philosophical Magazine for May 1869, Professor Syl- 

 vester gave as a deduction from a solution of an equation in 

 finite differences the following notable identity : — 



l = 1+1 >¥ + ? T - 3 3,4 



The origin of it was so peculiar, that to the reader it could not 

 but seem a desirable thing to establish a more general result, 

 and point out some connexion between it and previous special 

 results of the same kind. This has been done in one direction 

 by Mr. J. W. L. Glaisher, in a paper recently read before the 

 Mathematical Society of London, in which he transforms the 

 product 



KiX'+gX'+f:) 



into a series, this series into a continued fraction, and then points 

 out that, from the identity of the product and the continued 

 fraction, Professor Sylvester's result directly follows as a parti- 

 cular case. The object of the present short paper is to establish 

 another such line of relationship. 



C 1 {%— l) w_1 

 Consider the definite integral I x n -. .; ,. dec. By inte- 



J„ (*-nr +l J 



gration by parts we find that 



Jo Vff+1/ J (#+l) m+1 



* Communicated by the Author. 



