356 Mr. Lubbock on the Wave-surface 



If the planes, 



Vx-\- m' y-\- n'z = 

 which is parallel both to the resultant and to the direction of 

 displacement, and 



lx+ my+nz — O^ which is parallel to the plane of the 

 waves, are perpendicular to each other, 

 IV ■\- mm' -\- nil' ~ 

 {a^-c') cosXm {a^-h") cos Xn _ 

 ^+ (c2_ 6^) cos YZ "^ {b^-c^) cos Z I ~ 

 or, [a^—c^) m cos Xcos Z -f {b'^—a^) n cos Xcos Y 



4. [c'^-b'^)lcos YcosZ = 0. 

 This is the equation of Fresnel, Mem. de V Acad., vol. vii. 

 p. 115. line 3, and so far the reasoning is the same as his, the 

 only change being in the notation, which I have rendered 

 symmetrical by introducing the quantities /, »z, n instead of 

 1, B, C. 



This equation may be put in the form 

 «* cos X {m cos Z — n cos Y} +6^ cos Y [n cos X — Z cos Z} 

 + c^ cos Z{lcosY-m cos X} = (D.) 



This is the equation (A) of Fresnel, p. 113, and is the same 

 as Mr. Sylvester's equation (6.) L. and E. Phil. Mag., 1837, 

 vol. xi. p. 4<63. m cos 2 — n cos Yis identically equal to 

 cos Z {m cos Z—n cos Y} 

 cos Z 

 _ w {1 — cos^ X — cos^ Y} +1 cosXcosY + m cos^ X , ^ 

 "~ cos Y 



Similarly 



— w cos Xcos Y —/{ 1 — cos^ Y} 



n cos X — / cos Z — ii— — • 



cos Z 



Substituting these values of w cos I —n cos Yand n cos X 



— / cos Z in equation (D.), I get 



a^ cos X [m (1 — cos^ X) + / cos X cos Y} 

 •— h^ cos Y [l (1 — cos^ Y) + w cos X cos Y] 

 + c" cos Z [I cos Y — m cos X} = 0. 

 This equation may be put in the form 



- {a^ cos^ X + i^ cos^ Y + c^ cos^ Z} {m cos X - I cos Y] 



+ a^ m cos X — b"^ I cosY =: 0. 

 This equation is identical with the following equation of 

 Fresnel, obtained by differentiations, 



v^{u7i — ^m) =■ a^ an — b^ ^m, p. 1 30, or 



v^ {m cos X — Z cos Y) ■\- a^ m cos X — 6^ / cos Y = 0, 



