418 Mr. Lubbock on the JVave-surface in 



subject by eminent mathematicians ; perhaps however it may 

 not be uninteresting to return to Fresnel's reasoning through 

 which the equation to the wave-surface was originally esta- 

 blished, so as to facilitate the comparison of Fresnel's ideas 

 with those which have been since developed by M. Cauchy, 

 and by other philosophers*. With this view I have drawn up 

 the following very brief remarks. It seems to have been ge- 

 nerally considered that M. Cauchy's wave-surface is identical 

 with that of Fresnel, but I confess that I do not think that 

 this point has been sufficiently considered ; and indeed it has 

 already been asserted by Mr. Kelland that little construction 

 beyond the explanation of dispersion can be put upon M. 

 Cauchy's results, from their great complexity. 



Let the displacements of the particle m in the directions of 

 *>y>*he £,ij, ?, and those of m', ? + A £,3/ + At/, ? + A? at the 

 end of the time t 9 then 



d 2 £ r 



-~ m m2| (r<f> -f i>(r)Ax*} A£ t^rAj/A^Ar) 



-f- \J/r Ax Az A? \ 

 ?? = msf^rAyAxAg + [<p r + +(r Af} A>j 



-f tyr Ay A 2A5 r 



d 2 £ f 



— -J- — xt\ Si tyr Ax Az Ag + ^r Ay A z A»j 



dr L 



+ {$r + iKr)A**} A?}. 



These are M. Cauchy's equations, Exercises, vol. iii. p. 192: 

 they are also given by Mr.Tovey, L. & E.Phil. Mag., vol. viii. 

 p. 9; and by Mr. Kelland, Cambridge Trans., vol. vi. p. 158. 



The remarks of Fresnel, p. 85 to 95, Mem. de V Institute 

 vol. vii. amount to showing that these equations may, when 

 the axes of elasticity are taken for the coordinate axes, be re- 

 duced to the form 



i = m S <[<P r + * (r) A/ j^A 13 



!& =lti:{^ + ^(r)A^JA5. 



[• Prof. Powell's " Abstract of M. Cauchy's Views" appeared in Lond. 

 and Edinb. Phil. Mag., vol. vi. p. 16, et seg.- Edit.] 



a 



