736 
MH. J. LARMOR ON A DYNAMICAL THEORY OF 
and this must usually be either because the medium offers no resistance to laminar 
compression, or because it is incompressible, the case of rotational elasticity being 
however not thus restricted. The rays are most simply defined as the paths of 
the energy. 
23. Let us consider the first of these hypotheses, that of null velocity of 
longitudinal waves. At the interface the tangential components of the displacement 
must be continuous, otherwise there would be very intense tangential tractions 
acting in the thin interfacial layer of transition, such as could not be equilibrated b}" 
the tractions outside that layer. The normal components of the displacement need 
not be made continuous, for the neighbourhood of this thin interfaciai layer will 
stretch without effort as much as may be required. The tangential stresses must be 
continuous across the layer of transition, otherwise they would produce very great 
acceleration of this layer which could not be continuous with the moderate accelera¬ 
tions outside it. As we have thus already obtained the sufficient number of conditions 
the normal pressure need not also be explicitly made continuous, for the continuity of 
tangential displacements should secure its continuity as well ; if the medium is 
constituted so as to regularly reflect wmves at all, this must be the case, and it is clear 
on a moment’s consideration of the formula for the pressure that it is so in a labile 
medium of isotropic elastic-solid type. We have thus the four conditions, continuity of 
tangential displacement and of tangential stress; and the one sufficient condition 
which will secure that they also make the normal stress continuous, i.e. that the 
medium is a possible one, is that there shall be no loss of energy in the operation of 
reflexion and refraction. The four conditions here specified are mathematically 
equivalent to those of Fresnel’s theory of reflexion; and the satisfaction of the fifth 
condition carries with it the justification of that theory for the type of medium which 
it implies. For the case worked out by Fresnel, that of isotropic media, the 
constitution of his medium is thus limited to be precisely that of the labile oether 
of Lord Kelvin ; in order to satisfy also the fifth condition, that of continuity of 
energy, we are constrained to take the displacement perpendicular to the plane of 
polarization, which gives a reason independent of experiment for Fresnel’s choice. 
24. Let us next consider the second form of hypothesis, that of incompressibility. 
At the interface all three components of the displacement must now be continuous; 
and to obtain a solution, there is needed only one other condition, which may be taken 
to be the preservation of the energy of the motion. Here, as Neumann remarks, there 
is absolutely nothing assumed about the elastic condition of the media, which may in 
fact remain wholly unknown except as to their assumed incompressibility and as to the 
law of density, and the problem of reflexion will nevertheless be completely solved. 
But if w^e go further than this, and attempt to speculate about the elasticity of the 
optical medium, it must be limited to be of such nature as also to satisfy two other con¬ 
ditions which are involved in the continuity of the tangential stress at the interface. 
Thus on the principles that the energy is propagated along the rays, that it is at 
