220 EQUATIONS OF MONOCHROMATIC LIGHT IN MEDIA 



12. From the general equation given above (8, 12, or 15), in con- 

 nection with the solenoidal hypothesis, we may easily derive the laws 

 of the propagation of plane waves in the interior of a sensibly homo- 

 geneous medium, and the laws of reflection and refraction at surfaces 

 between such media. This has been done by Maxwell,* Lorentz,t and 

 others, J with fundamental equations more or less similar. 



The method, however, by which the fundamental equation has 

 been established in this paper seems free from certain objections 

 which have been brought against the ordinary form of the theory. 

 As ordinarily treated, the phenomena are made to depend entirely 

 on the inductive capacity and the conductivity of the medium, in a 

 manner which may be expressed by the equation 



(16) 



which will be equivalent to (12), if 



where K and C denote in the most general case the linear vector 

 functions, but in isotropic bodies the numerical coefficients, which 

 represent inductive capacity and conductivity. By a simple trans- 

 formation {see (9) and (10)}, this equation becomes 



e-'=* ,2 (is) 



4-7T Z7T 



where G' 1 represents the function inverse to 0. 



Now, while experiment appears to verify the existence of such a 

 law as is expressed by equation (12), it does not show that has the 

 precise form indicated by equation (16). In other words, experiment 

 does not satisfactorily verify the relations expressed by (16) and (17), 

 if K and C are understood to be the operators (or, in isotropic bodies, 

 the numbers) which represent inductive capacity and conductivity in 

 the ordinary sense of the terms. 



The discrepancy is most easily shown in the most simple case, when 

 the medium is isotropic and perfectly transparent, and reduces to a 

 numerical quantity. The square of the velocity of plane waves is 



r\ 



then equal to j , and equation (18) would make it independent of the 



* Phil. Trans. , vol. civ (1865), p. 459, or Treatise on Electricity and Magnetism, 

 chap. xx. 



t Schlomilch's Zeitschrift, vol. xxii, pp. 1-30 and 205-219 ; xxiii, pp. 197-210. 



: See Fitzgerald, Phil. Trans., vol. clxxi, p. 691; J. J. Thomson, Phil. Mag., 

 (5), vol. ix, p. 284 ; Rayleigh, Phil. Mag. (5), vol. xii, p. 81. 



That the electromagnetic theory of light gives the conditions relative to the boundary 

 of different media, which are required by the phenomena of reflection and refraction, 

 was first shown by Helmholtz. See Crelle's Journal, vol. Ixxii (1870), p. 57. 



