and Electromagnetic Theory. 331 



IL'Y'affi' which the fundamental equations require to be 

 continuous at a surface of separation. For the transfer of 

 energy in reflexion and refraction something more than the 

 continuity of V (X'/S'— Y'a!) is wanted, viz. the independence 

 of the flux formulae for the original and reflected waves 

 expressed by 



Y{X , ff-Y , a) l +Y{X f ^-Y'cc l -) 2 = Y(X'/3'-Y f cc , ) z . (34) 



the suffixes attaching to the original reflected and refracted 

 waves. If we set out from X a ' = X 1 / + X 2 / , and similar 

 equations for Y'a!/3\ the proof requires 



X/&' - YiW + X//V - Y 2 V = 0, 



or the vanishing of cross-products. Take the first medium 

 for convenience to be aether and apply (27) and (30) exactly 

 as in establishing (31). The quantity which should vanish is 



%*&{ (Vi^-tr) (1 -U 2 /V) + (Yn 2 -w)(l-JJ 1 IY) } 

 - KV-U072%«i + (V-U 2 ) 7l 2W 

 + (1 — 2/iJ 2 ) (722^^ + y 1 ^ua 2 ) 



+ y| (Yn 2 — w)%aa 1 Xl 1 a 2 + (Vn 1 — w)l t ua 2 Xk 0C l [ 5 



the work so far involving no special relations. The first line 

 vanishes by (21), which also makes 



(V-U 1 )24«l=(V-U J )(/ 1 a 1 +m lJ 8 l ) + (V-IIJn iyi 



=7,{(V-U,)n s -(V-U s )n l }, 



and the second line vanishes when the other part is similarly 

 treated. Again using (21), 



(Yn 2 — w)1,l x ot. 2 = y 2 \r n (Yn 2 — w) + n 2 (Yn x — w)} % 

 and 



l-Sy 2 =-2rt I « 3 + y (ni + n 2 ) 



= — y { ni(Yn 2 — w) + n^Y^ — w) \, 



so that the residue vanishes. 



The cross-products do therefore vanish in virtue of 2£X = 0, 

 2/a = 0, and. the geometrical laws of reflexion. 



Equation (31) written with the formula of interpretation 

 (32) for waves incident in aether on a moving dielectric is 



(Vn 1 -tr)S I +(V n ,-«c)S J = (^ '--,)S to . (35) 

 an equation for the continuity in the rates of arrival and 



