354! Mr. Lubbock on the Wave-surface 



t^^{r) sin^ (^) A y Ly\ = F'. 



The question is to show that the equations (B) and (C) 

 may be satisfied simultaneously by real values of the nine 

 quantities a, 6, &c., and to determine them. The proof of 

 this proposition is now, in consequence of the signification 

 which I have here assigned to the quantities D, E, F. D', E', F', 

 word for word the same as that which M. Poisson has given 

 of the existence of three principal axes of rotation in the Me- 

 moires de V Academie, vol. xiv. p. 320, and which of course it is 

 undesirable here to repeat. 



The kind of motion here supposed to take place, the 

 etherial molecule vibrating in a straight line in a plane touch- 

 ing the wave-surface, and making with the coordinate axes 

 the constant angles X, Y, %, is that, contemplated by Fresnel in 

 the reasoning which led him to the equation of the wave-surface. 

 Suppose the axis x to coincide in direction with the ray, 

 and the vibrations to be performed in planes at right angles 

 to that direction, then A | = ; and suppose 



>j = X { a sin (w t — kx)) 



}^ — ^{^s'm{nt — kx + b)} 



A >) = s/« [2 sin^ ( -^) sin (w 2f - ^ ^) 



— sin (^ Aa;) cos (w ^ — ^ ^) V 

 Af= SJ/sf 2sin2 [-^) sm{nt-kx + b) 



— sin (^Aa?) cos {nt — kx +b) V 

 2-|4'»"A«A.?/A*) J- 



= S I «^I/r2sin2 ^^A?) sin {nt-kx) Az Ay\ 

 — a \J/ r sin (^ A x) cos {nt — k x) A % A y\ 

 = sm{nt'-kx)'Z |«vKr)2sin^(^A£) A ^ A 3/ | 



-- cos [nt - kx) S {a\I/(r) sin(^A x) A z Ay}. 

 If the medium be constituted as before, p. S5S, line 8, the 

 second term obviously equals zero. 



If J/ = y cos 6 — « sin 6 z =■ z' cos s + y' sin s 



y' = y cos g + 2; sin e z' = z cose — y sin s. 



The quantity 1,1 u^ {r)2 s'm^ —^ A zAy\ = 0, if the 



