THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
731 
element of the interface to be parallel to the plane of yz, so that (I, m, 7*,) = (I, 0, 0) ; 
the surface integral term corresponding to one side of the interface is now 
dU 
snds, 
where By, B^ are perfectly arbitrary, subject only to being continuous across the 
interface. Thus to make the surface integral part of the variation vanish, we must 
have clUjclg and ddjjdh, the tangential components of the traction, continuous across 
the interface ; it follows from the first of the equations of motion that the continuity 
of ^ is also thereby secured, provided the density is the same on both sides ; and the 
normal traction on the interface is null. The continuity in the flow of energy across 
the interface is of course also necessarily involved. Of the complete set of six condi¬ 
tions only four are thus independent, which is the precise number required for the 
problem of optical reflexion between crystalline media. 
It has not been necessary to assume incompressibility of the medium in order to 
avoid waves of longitudinal disturbance. A medium of this type, however hetero¬ 
geneous in elastic quality from part to part, whether compressible or not, will 
transmit waves of transverse displacement in absolute independence of waves of 
compression, provided its density is everywhere the same; the one type of wave 
cannot possibly change into the other. 
16 . If 
V> 0 = curl (1^1, 7^1, Cl), 
so that 
if, g, h) - - V-(C'i, r/j, Cl), 
and if the equations of propagation are referred to the principal axes of the medium 
so that now 
they assume the form 
U = 1 {cdp -1- 6^ -f <r]d), 
(r- 
P di, ^i) = c®Ci), 
which are precisely Fresnel’s equations of crystalline propagation.^" The vector 
(^1, T7i, Cl) of Fresnel is at right angles to the plane of polarization, therefore its 
curl (C^, y, C) which is the displacement of the medium on MacCullagh’s theory, is 
in the plane of polarization. 
17 . In the theory of reflexion the tangential components of the displacement are 
continuous, and the tangential components of the stress are continuous, these 
conditions, or the more direct conditions of continuity of displacement and continuity 
* MacCullagh, ‘Proc. R.I.A.,’ vol. IL, 1841; ‘Collected Works,’ p. 188. 
5 A 2 
