THE ELECTRIC AND LUMINIFEROUS MEDIUM. 725 
there were six interfacial conditions to be satisfied, three of displacement and three of 
stress, while in the absence of compressional waves there were enough variables to 
satisfy only four of them; he cut the knot of this difficulty by assuming that the 
displacement must be continuous, to avoid rapture of the medium at the interface, 
and assuming that there is no loss of energy in the act of reflexion and refraction of 
the light, thus asserting the absence of waves of compression, and at the same time 
leaving the conditions as to continuity of stress altogether out of his account. As 
his displacement is in the plane of polarization, the solution arrived at by Neumann 
is formally the same as MacCullagh’s ; but it can be shown that the reasoning by 
which Neumann arrived at it, from the basis of an elastic solid aether, is invalid, so 
that the solution as stated by him must be considered to be the result of a fortunate 
accident, the correctness of which he would have had no real ground, in the absence 
of comparison with observations, for anticipating; while MacCullagh afterwards (in 
1839) placed his own empirical theory on a real dynamical foundation. 
10. The hypothesis on which Neumann’s surface-conditions are virtually based has 
been expounded and amplified in more recent times by Kirchhoef;^ and in this 
form it is often quoted as Kirchhoee’s principle. The analysis of Kirchhoee also 
amends Neumann^s defective energy-function by the substitution for it of the one 
determined by Green, by the condition that the displacements in two of the three 
types of waves that can travel unchanged in the medium are in the plane of the wave- 
front. About the rate of propagation of the third wave, involving compression in the 
medium, Kirchhoee makes no hypothesis, but he avails himself of the remark 
(originally due to MacCullagh) that the transverse waves involve no compression, 
and therefore are independent, as regards their propagation, of the term in the 
energy which involves compression. He assumes that in the act of reflexion and 
refraction no compressional waves are produced ; and that this is so because extra¬ 
neous forces act on the interface just in such manner as to establish the continuity of 
stress across it, while on account of the conservation of the energy they can do no 
v/ork in the actual motion of the medium at the interface. The explicit recognition 
of such forces constitutes Kirchhoee’s principle ; as to their origin he says that it lies 
in traction exerted by the matter on the aether which is unbalanced at the surface of 
discontinuity, and that they are somehow of the same nature as the capillary force at 
the interface between two liquids; as to their happening to be precisely such as will 
extinguish the compressional waves, he merely says that it must be so, because as a 
matter of fact no compressional waves are produced by the reflexion, the energy being 
assumed to be all in the reflected and refracted light-waves. On the other hand, the 
pure elastic theory has been worked out on Neumann’s hypothesis, for the simple 
case of an isotropic medium, without the assumption of these extraneous forces, by 
Lorenz, Lord Rayleigh, and others, and has been shown to lead to loss of light 
* G. Kirchhoff, “ Ueber die Reflexion and Bi’ecbung des Licbtes an der Grenze krystalliniscber 
Mittel,” ‘Abb. der Berl. Akad,,’ 1876; ‘ Ges. Abb.,’ p. 367. 
