AND THE THEORY OF A QUASI-LABILE ETHER 235 



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 (, and therefore by (6) 

 ty&, will be infinite. Now both (g and "SF are discontinuous at the 

 interface, but infinite values for "^(S are not admissible. Therefore 

 u' and v' are continuous. Again, if u or v is discontinuous, u f or v' 

 will become infinite, and therefore u* or v". 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 conditions, 

 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 w" that if u, v, u', and v' are continuous 

 at the interface w and w" will also be continuous. Now w" is equal 

 to the component of (: normal to the interface. The following 

 quantities are therefore continuous at the interface : 



the components parallel to the interface of (, \ 

 the component normal to the interface of "^(5, (7) 



all components of curl (5.J 



To compare these results with those derived from the electrical 

 theory, we may take the general equation of monochromatic light on 

 the electrical hypothesis from a paper in a former volume of this 

 Journal. This equation, which with an unessential difference of nota- 

 tion may be written * 



-Potg-VQ = 4ir$g, (8) 



was established by a method and considerations similar to those which 

 have been used to establish equation (6), except that the ordinary law 

 of electrodynamic induction had the place of the new law of elasticity. 

 ^ is a complex vector representing the electrical displacement as a 

 harmonic function of the time ; $ is a complex linear vector operator, 

 such that 4^$^ represents the electromotive force necessary to keep 

 up the vibration f. Q is a complex scalar representing the electro- 

 static potential, VQ the vector of which the three components are 



dQ d dQ, 



dx } dy 1 dz' 



Pot denotes the operation by which in the theory of gravitation the 

 potential is calculated from the density of matter.! When it is 



* See page 218 of this volume, equation (12). 



t The symbol - Pot is therefore equivalent to 47rV~ 2 , as used by Sir William Thomson 

 (with a happy economy of symbols) at the last meeting of British Association to express 

 the same law of electrodynamic induction, except that the symbol is here used as a 

 vector operator. See Nature, vol. xxxviii, p. 571, sub init. 



