FRESXEL ON DOUBLE REFRACTION. 303 



method because it appeared to be simpler than eUmination ; yet 

 nevertheless the calculations into which it led me are so long 

 and tedious that I do not think it advisable to give them here. 

 I shall content myself with saying that the condition expressed 

 by the equation (C.) is satisfied by the following equation : — 



- ^2 («2 + c2) Z/2 - C^ (a2 + ^2) ^2 _^ g2 ^2 ^2 = 0.. (D.) 



I had arrived at this equation by determining first the intersec- 

 tion of the surface of the wave with each of the coordinate planes, 

 an intersection which presents the union of a circle with an 

 ellipse. I remarked afterwards that a surface offering the same 

 character was obtained by cutting the ellipsoid by a series of 

 diametral planes, and drawing thi'ough its centre perpendicu- 

 larly to each plane radii vectores equal to half of each of the 

 axes of the diametral section ; for the surface which passes 

 through the extremities of all these radii vectores thus deter- 

 mined, gives also the union of a circle and an ellipse in its inter- 

 section with the three coordinate planes ; it is moreover of the 

 fourth degree only, and the identity of the sections made by the 

 three rectangular conjugate diametral planes in these two sur- 

 faces would have been to me a sufficient proof of their identity 

 if I had been able to demonstrate that the equation of the wave 

 could not surpass the fourth degree, a result which seemed to 

 follow from the conditions themselves of its generation ; since 

 there are only two values for the square (u^) of the distance of 

 the origin from the tangent plane, so that the surface cannot 

 have more than two real sheets ; but as it was not impossible 

 that the equation sought might contain besides imaginary sheets 

 [nappes), it was necessary to obtain direct proof, as I have done, 

 that the equation of the fourth degree, to which the ellipsoid 

 had conducted me, satisfied equation (C), which expresses the 

 generation of the surface of the wave. 



Very simple process which leads from the equation of an 

 ellipsoid to that of the wave surface. 



The calculation by which I arrived at equation (D.) is so simple, 

 that 1 think it right to give it here. 



I take an ellipsoid which has the same axes as the surface of 

 elasticity ; its equation is 



b' c" .x-2 + c2 c' y^ + a" 6^ z' = a^ ^2 ^.q. 



