DOUBLE REFRACTION. 121 



results arrived at by Fresnel. The axes of this ellipsoid coincide 

 in direction with, and are inversely proportional to, the axes of 

 Fresnel's generating ellipsoid; and Mr. M'Cullagh has demon- 

 strated the truth of Fresnel's construction for the wave-surface, by 

 means of a simple geometrical relation between its tangent planes 

 and the sections of the two ellipsoids. 



In the third supplement to his " Essay on the Theory of Systems 

 of Rays"* Professor Hamilton has presented that portion of Fres- 

 nel's theory, which relates to the fundamental problem of the 

 determination of the velocity and polarization of a plane wave, in 

 a very elegant analytical form ; and from the velocity and direction 

 of the wave he deduces those of the ray, and therefore the form of 

 the wave-surface, by means of the general relations suggested by 

 his view of mathematical optics. 



In this system, of which the author gave a brief sketch at the 

 late meeting of the Association, the laws of reflexion and refraction, 

 ordinary or extraordinary, are comprised in two fundamental ex- 

 pressions, which state that the partial differential coefficients of the 

 first order of a certain function taken with respect to two final 

 coordinates in the plane which touches the reflecting or refracting 

 surface at the point of incidence, are not altered by reflexion or 

 refraction. The function here considered is the characteristic func- 

 tion of the author, whose particular form may be considered as 

 characterizing the optical system, and on whose properties, he finds, 

 all the problems of mathematical optics may be made to depend. 

 On the principles of the wave-theory, this function is equal to the 

 undulatory time of propagation of light, from any one assumed 

 point to another, in the same or in a different medium ; and the 

 expressions just alluded to signify simply that the components of 

 normal slowness of the wave parallel to the bounding surface, or the 

 reciprocals of the velocities of wave-propagation resolved in the 

 direction of that surface, are not changed by reflexion or refraction. 

 The normal slowness of wave-propagation is, then, of fundamental 

 importance in this theory ; and if it be represented in magnitude 

 by a line taken in its direction, there is obtained for its expression 

 a curved surface which, on the principles of Fresnel, is found to be 

 a surface of two sheets, connected with the wave-surface by a 

 remarkable relation of reciprocity. When this relation is com- 



* Transactions of the Royal Irish Academy, vol. xvii. 



