THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
735 
The same end might have been attained by taking [f, g, li) to denote displacement 
and (^, rj, Vj proportional to rotation in the variational equation; for V~ g, Q 
= — curl {/, g, h), and the operator may be replaced by a constant so far as 
regards light-propagation in a single medium. This interchange, which has already 
been indicated in § 18, does not affect the development of the variational equation 
except as regards surface-integral terms ; and the character of the modification of the 
geometrical relations of the wave surface, on passing from the one theory to the 
other, is now open to inspection.* 
[21. (Added June 14.) The formal relations between these various mechanicai 
theories may be very simply traced by comparing them with the electromagnetic 
scheme of Ma.x\vell. In that theory the electric and magnetic inductions, being 
circuital, are necessarily in the plane of the wave-front; while the electric and mag¬ 
netic forces need not be in that plane. On taking the electric or the magnetic 
induction to represent the mechanical displacement of the medium, the electric theory 
coincides formally with that of Feesnel or that of MacCdllagh respectively; while 
on taking the electric or the magnetic force to represent the mechanical displacement, 
we obtain the equations of the correlative theories of Boussinesq, Lord Kelvin, and 
other authors.! Thus, for example, it follows at once from this correlation that the 
combination of seolotropic inertia with labile isotropic elasticity wiU lead, not only to 
Fresnel’s wave surface as Glazebrook has shown, but also to MacCullagh’s theory 
of crystalline reflexion and refraction. If we suppose the magnetic cjuality of the 
medium to take j^art in the vibrations, as would probably be the case to some extent 
with very slow electric waves, the ecjuations of projiagation would possess features 
analogous to those due to an alteration of density in passing from one medium to 
another, on the mechanical theory here adopted. But the continuity of normal dis¬ 
placement of the medium could not now be satisfied in the problem of reflexion, the 
appropriate magnetic condition being instead continuity of induction. A homogeneous 
mechanical medium representing or illustrating such a case would thus have to possess 
suitable labile properties; in the ordinary optical circumstances in which magnetic 
quality is not effective, the degree of compressibility is on the other hand immaterial, 
and no normal wave will be started in reflexion.] 
Treatment of the Problem of Reflexion hy the Method of Rays. 
22. We are now in a position to compare the various investigations of the problem 
of reflexion, by means of rays, that have been given by Fresnel, Neumann, 
MacCullagh and others. It is a cardinal principle in all theories of transparent 
media that there is no loss of energy in the act of reflexion and refraction. 
Consequently there is no energy carried away by longitudinal waves in the sether; 
* Cf. J. Willard Gibbs, “A comparison of the electric theory of light and Sir W. Thomson’s 
theory of a qnasi-lahile sether,” ‘Phil. Mag.,’ 1889. 
t Cf. Drude, ‘ Gottinger Nachrichten,’ 1892. 
