of Light and the Theory of a Quasi-labile ^ther. 241 



For let u, v, to be the components of (i, 



w' „ „ „ curl (g, 



lo" „ „ „ curl curl (g, 



so that 



' ) <-' ) 



, __ dio dv f_du dio f_dv du 



~~ dy dz^ dz dx^ dx dy^ 



f, dw' dv' f,_ du' dw' ,, _ dv' du' _ 



dy dz- dz dx^ dx dy ' 



and let the interface be perpendicular to the axis of Z. It is 

 evident that if u' or v' is discontinuous at the interface, the 

 value of u^' or v" becomes in a sense infinite, i. e. curl curl (S, 

 and therefore, bj (6), "^g will be infinite. Now both (§, and 

 "9 are discontinuous at the interface, but infinite values for 

 "^d: are not admissible. Therefore w' and v' are continuous. 

 Again, if u or v is discontinuous, u^ or v' will become infinite, 

 and therefore u" or tJ'. Therefore u and v are continuous. 

 These conditions may be expressed in the most general manner 

 by saying that the components of @ and curl (§. parallel to the 

 interface are continuous. This gives four complex scalar con- 

 ditions, or in all eight scalar conditions, for the motion at the 

 interface, which are sufficient to determine the amplitude and 

 phase of the two reflected and the two refracted rays in the 

 most general case. It is easy, however, to deduce from these 

 four complex conditions two others, which are interesting and 

 sometimes convenient. It is evident from the definitions of 

 w' and id' that, if u, v, u', and v' are continuous at the inter- 

 face, v/ and lo" will also be continuous. Now —w" is equal 

 to the component of '^^ normal to the interface. The fol- 

 lowing quantities are therefore continuous at the interface : — 



the components parallel to the interface of ©, "^ 



the component normal to the interface of ^Q:, V (7) 



all components of curl ®. ) 



To compare these results with those derived from the elec- 

 trical theory, we may take the general equation of mono- 

 chromatic light on the electrical hypothesis from a paper in a 

 former volume of Silliman's American Journal. This equa- 

 tion, which with an unessential difference of notation may be 

 written* 



-Pot§-VQ = 47r$g, .... (8) 

 was established by a method and considerations similar to those 



* See the Amer. Joum. Sci. yoI. xxv. p. 114, equation (12). 

 Phil. Mag. S. 5. Voh 27. No. 166. March 1889. R 



