AND THE THEORY OF A QUASI-LABILE ETHER. 237 



In other words, the displacements in either theory are subject to the 

 same general and surface conditions as the forces required to maintain 

 the vibrations in an element of volume in the other theory. 



To fix our ideas in regard to the signification of "SF and 3?, we may 

 consider the case of isotropic media, in which these operators reduce 

 to ordinary algebraic quantities, simple or complex. Now the curl 

 of any vector necessarily satisfies the solenoidal condition (the 

 so-called " equation of continuity "), therefore by (6) ^@ and @ will 

 be solenoidal. So also will ^ and <3?3f in the electrical theory. Now 

 for solenoidal vectors 



(15) 



so that the equations (6) and (10) reduce to 



For a simple train of waves, the displacement, in either theory, may 

 be represented by a constant multiplied by 



e 4gt+ax+by+cz) 



Our equations then reduce again to 



(19) 

 (20) 



Hence 



a 2 

 \ff - 1 _ ;*. _ & 



The last member of this equation, when real, evidently expresses the 

 square of the velocity of light. If we set 



-, (22) 



k denoting the velocity of light in vcacuo, we have 



n 2 = k 2 V = k 2 $-\ (23) 



When n 2 is positive, which is the case of perfectly transparent 

 bodies, the positive root of n 2 is called the index of refraction of the 

 medium. In the most general case, it would be appropriate to call 

 n or perhaps that root of n 2 of which the real part is positive the 

 (complex) index of refraction, although the terminology is hardly 

 settled in this respect. A negative value of n 2 would represent a 

 body from which light would be totally reflected at all angles of 

 incidence. No such cases have been observed. Values of n 2 in which 

 the coefficient of i is negative, indicate media in which light is 



