264 REPORT—1862. 
from pressure is first considered*. Green shows that the ellipse which is the 
section of the ellipsoid of elasticity by a diametral plane, parallel to the wave’s 
front, if turned 90° in its own plane, belongs to a fixed ellipsoid, which gives 
at once Fresnel’s elegant construction for the velocity of propagation and 
direction of the plane of polarization; but it is necessary to suppese that in 
polarized light the vibrations are parallel, not perpendicular, to the plane of 
polarization. 
The general case in which the medium is not assumed to be symmetrical with 
respect to three rectangular planes, and in which therefore ¢ contains twenty- 
one arbitrary constants, is afterwards considered; and it is shown that the 
hypothesis of strict transversality leads to fourteen relations between them, 
leaving only seven constants arbitrary. But the function obtained on the 
assumption of planes of symmetry contains no fewer, for the four constants 
relating to these planes would be increased by three when the medium was 
referred to generalaxes. Hence therefore the existence of planes of symmetry 
is not an independent assumption, as in Cauchy’s theory, but follows as a 
result. 
In this beautiful theory, therefore, we are presented with no forced rela- 
tions like Cauchy’s equations; the result follows from the hypothesis of 
strictly transversal vibrations, to which Fresnel was led by physical considera- 
tions. The constant , remains arbitrary, and it is easy to see that this 
constant expresses the square of the velocity of propagation of normal vibra- 
tions. Were this velocity comparable with the velocity of propagation of 
transversal vibrations, theory would lead us still to expect normal vibrations 
to be produced by light incident obliquely, though not by light incident 
perpendicularly, on the surface of a crystal, and the theory would still be 
exposed to many of the objections which have been already brought forward. 
But nothing hinders us from supposing, in accordance with the argument 
contained in Green’s former paper, that p is very great or sensibly infinite, 
which removes all the difficulty, since the motion corresponding to this term 
in the expression for —2 % would not be sensible except at a distance from 
the surface comparable with the length of a wave of light. Hence, although 
it might be said, so long as x was supposed arbitrary, that the supposition of 
rigorous transversality had still something in it of the nature of a forced 
relation between constants, we sec that the single supposition of incompressi- 
bility (under the action of forces at least comparable with those acting in the 
propagation of light)—the original supposition of Fresnel—introduced into 
the general equations, suffices to lead to the complete laws of double refrac- 
tion as given by Fresnel. Were it not that other phenomena of light lead us 
rather to the conclusion that the vibrations are perpendicular, than that they 
are parallel to the plane of polarization, this theory would seem to leave us 
nothing to desire, except to prove that we had a right to neglect the direct 
action of the ponderable molecules, and to treat the ether within a crystal as 
a single elastic medium, of which the elasticity was different in different 
directions. 
In his paper on Reflexion, Green had adopted the supposition of Fresnel, 
that the vibrations are perpendicular to the plane of polarization. He was 
naturally led to examine whether the laws of double refraction could be 
explained on this hypothesis. When the medium in its undisturbed state is 
exposed to pressure differing in different directions, six additional constants 
are introduced into the function ¢, or three in case of the existence of planes 
* The results obtained for this case remain the same if we suppose the medium in its 
undisturbed state to be subject to a pressure alike in all directions, 
