Electrodynamics for Moving Ponderable Media. 55 



Further, we must transform from differential coefficients 

 with respect to a? l5 3/1, ~i, t 1} to those with respect to the 

 moving axes x\ y',z', t' by combining equations (1) and (13), 

 so that 



a} , =€~ 1 x 1 , y'=yi, z' = z u t'^efa + wax), 

 whence 



3 -1 3 , 3 3^ _ 3 



^i" e 3^ +e "V' 3*i~ 3*" 



After some algebraic work, equation (16) transforms into 



D-E = (K'-1)E' + ^[l',^] + [J'V' ) E']+^[jX B J-'] • (18) 



where 



K f -l=(l,e 2 ^ 2 )(K 1 -l), L' = (e, 1, 1)1* J'^e^. 



The second and third equations in the above vector 

 equation give 



D z -E z =(K;-i)E;H-^L;^f'+j/^+^j; w ^', 



Therefore finally, on substituting in (15) for D y — Ey and 

 D r — E^r, we have 



P~fy7 y ~ («* + K/-l)]E^ = »E JJ*'e*(l-w»-»V) -L,VV] 



|- (l-AoVy _ (e2 + K; _ 1) -| Eoi= -^[J^l-^-^-I^VV] 



If we take the case of a uniaxial crystal and let its axis of 

 rotation coincide with the axis of a?', i. e. with the direction 

 of motion of the medium, K y ' = K^' = K, and the equation to 

 determine V becomes 



T fi 1 -^- 10 ^ 2 _( 6 2 + K-lj] = +s[J x '6*(l-w*-wY) -€ 2 IVV] . (19) 



We note in passing that if J x ' = L/ = 0, V is determined 

 from the quadratic 



(K-l)V 2 +(V + w;) 2 =:l*, 



giving Fresnel's coefficient as the term of the first order 



* Putting V=U-w, this gives (K-1)(U— ic) 2 +TJ 2 =l, agreeing 

 with Larmor, ' iEther and Matter,' p. 59, U being the absolute velocity of 

 wave, and V its velocity relative to the medium, which is moving with 

 a velocity w. 



