158 Mr Glazebrook, On some equations connected with [Nov. 28, 



Put Ix + my + nz — Vt = co and let the displacement be S, some 

 function of a>, so that 



f=\S, 



g = v>s, 



h = vS. 



Then 



l\+ tn/j, + nv — (11), 



or the disturbance is in the wave front, and 



\ { 2 V-a 2 (m 2 + n 2 )} + fiflm + v?nl = 0, 



\a 2 Im + fi{V 2 - b 2 (n 2 + I 2 )} + Vc 2 mn = 0, 



\a 2 nl + n&mn +v{V 2 - c 2 (I 2 + m 2 )} = ; 



therefore X ( F 2 - a 2 ) + I \a 2 TK + b'mp + c'nv] = 0, etc. ; 



I X 



therefore ;=— - = -^— , etc. 



a 2 - V 2 of l\ + b -m/j, + c 2 nv 



Multiply by /, m, n respectively, and add 



I 2 m 2 n 2 



^-V 2 + b 2 -V' 2+ ?-V 2 . 



l\ + mfi + nv _ 



d 2 l\ + b 2 m/jb + c 2 nv 



Thus the relation between the direction of propagation and the 

 velocity of the wave is that given by Fresnel. 



Moreover, eliminating from the three equations the quantities 



V 2 and o?l\ + b 2 m/j, + c 2 nv, we get 



I - on - - n - 



X$ 2 - c 2 ) + -(c 2 -a 2 ) +~ (a 2 - If) = (13). 



Thus the direction of electric displacement is also given by 

 Fresnel's construction. 



These conclusions have already been arrived at by Maxwell, 

 Electricity and Magnetism, Vol. II. § 798, but he has given the 

 equations satisfied by F, G, H. In problems connected with the 

 reflexion and refraction of the waves the conditions are expressed 

 more easily in terms of f, g, It, and so it became necessary to form 

 the equations these quantities satisfy. 



We proceed next to the equations satisfied by the components 

 of magnetic induction. 



We have a, b, c, being these components : 

 da _ \ 1 dg 1 dh) 

 Tt-^WJ-z-I^dy]' 



