ON DOUBLE REFRACTION. 265 
of symmetry to which the medium is referred. For waves perpendicular to 
the principal axes, the directions of vibration and squared velocities of 
propagation are as follows :— 
Green assumes, in accordance with Fresnel’s theory, and with observation 
if the vibrations in polarized light are supposed perpendicular to the plane of 
polarization, that for waves perpendicular to any two of the principal axes, and 
propagated by vibrations in the direction of the third axis, the velocity of pro- 
pagation is thesame. This gives three, equivalent to two, relations among the 
constants, namely, 
A—L=B—M=C—N=y suppose, (9) 
which are equivalent to Cauchy’s equations (3). The conditions that the 
vibrations are strictly transversal and normal respectively do not involve the 
six constants expressing the pressures in equilibrium, and therefore remain the 
same as before, namely (7). Adopting the relations (7) and (9), Green proves 
that for the two transversal waves the velocities of propagation and the azimuths 
of the planes of polarization are precisely those given by the theory of Fresnel, 
the vibrations in polarized light being now supposed perpendicular to the plane 
of polarization. 
As to the wave propagated by normal vibrations, the square of its velocity 
of propagation is easily shown to be equal to 
; p+AP+ Bm? + Cn’ ; 
and as the constant p does not enter into the expression for the velocity of pro- 
pagation of transversal vibrations, the same supposition as before, namely that 
the medium is rigorously or sensibly incompressible, removes all difficulty arising 
from the absence of any observed phenomenon answering to this wave. 
The existence of planes of symmetry is here in part assumed. I say in part, 
because Green shows that the six constants, expressing the pressures in 
equilibrium, enter the equation of the ellipsoid of elasticity under the form 
K («*+y’+2*), where K is a homogeneous function of the six constants of the 
first order, and involves likewise the cosines 7, m,n. Hence the directions of 
vibration are the same as when the six constants vanish; the velocities of 
propagation alone are changed; and as the existence of planes of symmetry 
for the case in which the six constants vanish was demonstrated, it is only 
requisite to make the very natural supposition that the planes of symmetry 
which must exist as regards the directions of vibration, are also planes of 
symmetry as regards the pressure in equilibrium. 
We see then that this theory, which may be called Green’s second theory, 
is in most respects as satisfactory (assuming for the present that Fresnel’s 
construction does represent the laws of double refraction) as the former. I 
say in most respects, because, although the theory is perfectly rigorous, like 
the former, the equations (9) are of the nature of forced relations between 
the constants, not expressing anything which could have been foreseen, or 
