104 Prof. Powell on the Demonstration o/FresnePs Formulas 



and of volume dv\ k the real specific heat of the substance; 

 p the pressure under which it expands, so that pdv is the external 

 work performed by the expansion. The various expressions for 

 this equation which have appeared in the writings of Messrs. 

 Clausius and Thomson, and in my own, merely differ in form, 

 and a'*e all substantially equivalent to each other. It aflfords the 

 solution of every conceivable question where the mutual relations 

 of heat and of work by cubic expansion are concerned, and has 

 been abundantly and rigorously verified by experiment ; and I 

 think I am justified in maintaining it to be clear and simple as 

 well as general. 



I trust it will be understood, that, in making this statement, 

 I have no wish to detract from the merits of M. Hoppe, my sole 

 object being to defend the existing theory against the prejudicial 

 inference which might be drawn from the opening remark of the 

 English abstract of M. Hoppers paper — a remark which does not 

 appear in the original German. 



I have the honour to be, Gentlemen, 



Your most obedient Servant, 



Glasgow, July 2, 1866. W. J. Macquorn Rankine. 



P.S. I may take this opportunity of giving increased publicity 

 to a peculiar transformation of the equation A, which is useful 

 in certain special investigations. The paper in which it was 

 first given was read to the Royal Society of Edinburgh in 

 February 1855, but has not yet been published. 



M,={n,^-^y-r{,4-,,4^fi^ . (B) 



Po) %) '^0 denote the pressure, volume, and absolute temperature 

 of the substance in the ideal state of perfect gas. — W. J. M. R. 



XIII. On the Demonstration o/Fresnel's Formulas for Reflected 

 and Refracted Light ; and their Applications. — Part II. By the 

 Rev. Baden Powell, M.A., F.R.S, ^c, Savilian Professor 

 of Geometry in the University of Oxford*. 



. TN a former paper (see Phil. Mag. &c. July 1856) I have 

 J- placed in a connected point of view the several principles 

 and deductions leading to the well-known formulas of Fresnel, 

 as well as to certain modifications of them, for the amplitudes 

 .of the vibrations of the incident, reflected, and refracted rays, 

 vvhether polarized parallel or perpendicular to the plane of inci- 

 dence. I have also remarked on the question which has so long 



Communicated by the Author. 



