116 J. W. Gibbs—Equations of Monochromatic Light 
different from (12), since A and g are only particular values of 
and 
12. From the general equation given above (8, 12, or 15), in 
connection with the solenoidal hypothesis, we may easily 
derive the laws of the propagation of plane waves in the inte- 
rior of a sensibly homogeneous medium, and the laws of reflec- 
tion and refraction at surfaces between such media. This has 
been done by Maxwell,* Lorentz,+ and others,t with funda- 
mental] 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 con- 
ductivity of the medium, in a manner which may be expressed 
by the equation 
{So fC a\ Ax’ 
[Ulave —— (=- An? +) (S Pot [U ave oh rlalave), (16) 
which will be equivalent to (12), if 
K pC\ (47° 
An Qn 
where K and C denote in the most general case the linear 
vector functions, but in isotropic bodies the numerical coéfii- 
cients, which represent inductive capacity and conductivity. 
By a simple transformation [see (9) and (10)], this equation 
becomes ; 
ts a pe 18 
ce 4m ont’ (38) 
where 6" represents the function inverse to 8. 
ow, while experiment appears to verify the existence ° 
such a law as is expressed by equation (12), it does not show 
that @ has the precise form indicated by equation (16). Io 
other words, experiment does not satisfactorily verify the rela- 
tions expressed by (16) and (17), if K and © are understood to 
be the operators (or, in isotropic bodies, the numbers) which 
represent induction capacity and conductivity in the ordinary 
sense of the terms. 
a Phil. Trans., vol. cly (1865), p. 459, or Treatise on Electricity and Magnetism, 
ap. * 
+ Schlémilch’s Zeitschrift, vol. xxii, pp. 1-30 and 205-219; xxiii, pp. 197-210. 
¢ See Fitzgerald, Phil. Trans., vol. clxxi, p. - J. J. Thomson, Phil. Mag+ 
V, vol. ix, p. 284; Rayleigh, Phil. M 
: 
and refraction, was first shown by Helmholtz. See Crelle’s Jourual, vol. 
(1870), p. 57. 
