352 Mr. Lubbock on the Wave-surface 



reasoning of Fresnel, by which the equation to the wave- 

 surface in the theory of double refraction was first established, 

 and to facilitate the comparison of Fresnel's ideas with those 

 which have been since developed by M. Cauchy and by other 

 philosophers. 



As Prof. Powell has recently referred to this subject in a 

 communication read at the last meeting of the British Asso- 

 ciation, I wish to offer some remarks with respect to the ex- 

 istence of axes of elasticity'^, by which I mean axes such that 

 when they are taken for the direction of the coordinate axes 

 X, y, z, the equations of motion oftheetherial molecule m are 

 of the form 



^ = mS{<f>r+rl/(r)A^^} A^ 



(the notation being the same as in my former paper) or in the 

 words of Fresnel, which is the same thing, upon the existence 

 of " trois directions rectangulaires, suivant lesquelles, tout 

 petit deplacement de ce point, en changeant un peu les forces 

 auxquelles ilestsoumis, produitune resultante totale dirigee 

 dans la ligne meme de son deplacement." I conceive that the 

 reasoning of M. Cauchy in the Nouveaux Exercises^ p. 11, is 

 suffijcient to show that such axes generally exist at any point, 

 although I do not recollect that he has anywhere enunciated 

 the proposition precisely in the same shape as Fresnel. This 

 theorem bears a remarkable analogy to that of the existence of 

 three principal axes of rotation in Mechanics ; and to the other 

 theorem, also well known, that by a proper choice of coordi- 

 nates, the general equation to the curve surface of the second 

 order in x, ?/, and ;£ may always be so reduced as not to con- 

 tain the products x y, x z, y z multiplied by constant coeffi- 

 cients. Demonstrations of tliese theorems have been often 

 given, and may be found in the Memoii-e sur V attraction d'un 

 Ellipsoide homogene, by M. Pois^on, and in his Memoir e sur 

 le mouvement d'un corps solide. The same proof mutatis mu- 

 tandis is applicable to both those questions, and also to that 

 of the existence of axes of elasticity. 



Suppose 



A0 = ApcosX A>j = Ap cos Y A ^ =: A /s cos 21 

 p = A cos {n t — Jcr) 



Ap — — 2 A sin^ {"q" ) ^°^ {nt — k r) 

 + A sin [kAr) sin {nt — kr). 

 In order to prove the existence of axes of elasticity, it is 

 • See Fresnel's Memoire sur la Dcruble Refraction, p. 94. 



