atid Electromagnetic Theory. 329 



Also 



"C 1 - ^H" h tu *'> and K'" $ = ° • (28) 



with similar properties for X, since Xuct. = 1uu' and 

 2«X=2tiX / . Thus the letters with dash denote quantities 

 perpendicular to the ray, not to the wave-normal ; also since 



they are not mutually perpendicular, except for first-order 

 work in (uvio). From (27), 



mw»m(i- ^)s« 2 =k(i- ^)sx 2 =ksxx' 



-|m'+fW = B=B.(l-^);. . (29) 



this giving a definition of S. 



Again we may express at! in terms of the electric induction; 

 thus 



V*' = V* + ~JicY-vZ\ = —(mZ-nY) + ^JwY-vZ\ 



= K ((?-J) z -c|-5) Y } ; 



and similarly ' r-(30) 



™-*{(?-?)»-(t-?W' J 



relations which involve components of ray-velocity. From 

 (27) and (HO), 



V(X'/3'-YV) 



-(?-3(->=(?-J)s w 



From (»), nS -V.), ?, E (32) 



