li' On the Transformation of Optical Wave- Surf aces. [Jan. 18, 



moment. Start from the last equation (58). Hamilton's formula 

 (51) makes it become 



- [cJ/iH = VyVcvcH. (59) 



The elementary formula in vector algebra, 



VaVbc = b (ca)-c (ab), (60) 



transforms (59) to 



- [c>H = cv ( VcH) - ( vcv) cH, (61 ) 



or (vcv) c [c] /t-T- H = cv (VcH), (62) 



etc J 



from which 



r /P~i i 



(63) 



So far we have merely a changed form of the characteristic. But 

 the induction /tH is circuital. Therefore, taking the divergence of 

 (63), we obtain 



tJ2 -i _i 

 (Vcv)c-[c]/t^l cv(vcH), (64) 



or, which is the same, 



= V (VCV>- 1 -M c- 1 "^ (VcH). (65) 



Here vcH is the divergence of cH. It is the same as (cv)H. 



Now (65) only differs from the velocity equation (for plane waves) 

 in containing v instead of the unit normal N and d?ldt? instead of tr, 

 v being the wave-velocity. Thus, let 



eP 



then we shall have v 2 v 2 H = H, 



air 



where, however, v a is specialized, being only v 2 or cPfdz*. We 

 therefore put t^Va 2 for d?ldf* and Nv 3 for v in equation (65), thus 

 making 



= N Vs (N V3 cNv 3 ) ^-- [c]c-Vv s 2 Nvs (N Vs cH) (66) 

 We may now cancel out all the Vs's except the last, making 



= N [(NcN) /1 -'- [c]c-v] "'NrNvjcH). (67) 



