ON DOUBLE REFRACTION. 255 
suppose that in polarized light the vibrations are parallel, not perpendicular, 
to the plane of polarization. 
Let 1, m, n be the direction-cosines of the wave-normal. In the theory of 
Cauchy and Neumann, the square v* of the velocity of propagation is given 
by a cubic of the form 
v+a,v+a,v°+a,=0, 
where «,, a,, @, are homogeneous functions of the Ist order as regards 
L, M, N, P,Q, R, and homogeneous functions of the orders 2, 4, 6 as regards 
l,m, , involving even powers only of these quantities. For a wave perpen- 
dicular to one of the principal planes, that of y z suppose, the cubic splits 
into two rational factors, of which that which is of the first degree in v”, 
namely, 
v—m R—n? Q, 
corresponds to vibrations perpendicular to the principal plane. This is the 
same expression as results from Fresnel’s theory, and accordingly the section, 
by the principal plane, of one sheet of the wave-surface, which in this theory 
is a surface of three sheets, is an ellipse, and the law of refraction of that ray 
which is polarized perpendicularly to the principal plane agrees exactly with 
that given by the theory of Fresnel. 
For the two remaining waves, the squared velocities of propagation are 
given by the quadratic 
(v?—m* M—n’* P) (v?—m? P—n? N)—4in? n? P?=0; «2... (1) 
but according to observation the ray polarized in the principal plane obeys 
the ordinary law of refraction. Hence (1) ought to be satisfied by v?—(m? 
+n*) P=0, which requires that (M—P) (N—P)=4P?, on which supposition 
the remaining factor must evidently be linear as regards m?, n?, and therefore 
must be 
v—n? M—n? N, 
since it gives when equated to zero v?=M, or v?=N for m=1, orn=1. And 
since the same must hold good for eack of the principal planes, we must have 
the three following relations between the six constants, 
(M—P) (N—P)=4P*; (N—Q) (L—Q)=4Q’; (L—R) (M—R)=4R?... (2) 
_ The existence of six constants, of which only three are wanted to satisfy 
the numerical values of the principal velocities of propagation in a biaxal 
crystal, permits of satisfying these equations; so that the law that the ray 
polarized in the plane of incidence, when that is a principal plane, obeys the 
ordinary law of refraction is not inconsistent with Cauchy’s theory. This 
simple law is, however, not in the slightest degree predicted by the theory, 
nor even rendered probable, nor have any physical conditions been pointed 
out which would lead to the relations (2); and, indeed, from the form of 
these equations, it seems hard to conceive what physical relations they could 
express. Hence an important desideratum would be left, even if the theory 
were satisfactory in all other respects. : 
- The equation for determining v* virtually contains the theoretical laws of 
double refraction, which are embodied in the form of the wave-surface. The 
wave-surface of Cauchy and Neumann does not agree with that of Fresnel, 
except as the sections of two of its sheets by the principal planes, the third 
sheet being that which relates to nearly normal vibrations. Nevertheless the 
first two sheets, being forced to agree in their principal sections with Fres- 
nel’s surface, differ from it elsewhere extremely lttle. In Arragonite, for 
instance, in a direction equally inclined to the principal axes, assuming Rud- 
