MR. R. T. GLAZEBROOK ON PLANE WAVES IN A BIAXAL CRYSTAL. 291 
the effects they would produce, and which, though small, may be sufficient to constitute 
the difference between Fresnel’s and some rival theory. Moreover, the values of the 
principal refractive indices in topaz are so nearly equal that any conceivable wave- 
surface will differ but little from two spheres. 
To estimate then the weight which we may attach to these results, let us see how 
tar the experiments are consistent with one of the rival theories. 
Now the equation to the surface of wave slowness in the theories of Green and 
Cauchy involves more constants than the three principal refractive indices of the 
crystal. 
To determine these we must make the principal sections agree with experimental 
results in more points than the extremities of the axes; and it is probable, therefore, 
that they will differ but little throughout. 
In fact, the existence of these unnecessary constants is a radical defect in both these 
theories. 
Lord Rayleigh, however, has proposed a theory in which the only constants are 
the principal velocities (Phil. Mag., vol. 41, series 4, 1871). 
The shape of the surface of wave slowness is also considerably different from that 
given by Fresnel’s theory. 
Its principal sections consist of a circle and the inverse of an ellipse instead of a 
circle and an ellipse. 
He supposes “ the density of the ether in a crystal to be a function of the direction 
of displacement,” while the forces called into play are the same as in a homogeneous 
medium. 
From this he deduces as the equation to determine the velocity of a wave front, of 
which the direction cosines of the normal are l, m, n, the equation 
crZ 3 bhn 2 cV 
V 2 —« 2 + V T -b 2 + V 2 —c 3 “ 0 
a, b, c being the principal velocities. 
Let us find the velocity for a wave perpendicular to the plane zx, making an angle 
a with Ox, l —— sin a, m=0, n— cos a. 
V~((r cos^ a+or siir a) = crcr 
or since a= 1, c 3 = 1 — y 
V 3 {(l — y) COS 3 a-f- sin 3 a} = 1— y 
V 3 = (1 —y)(l — y cos 3 a) -1 
V =( 1 -ir)( 1 +iy cos2 a ) 
neglecting y 3 , &c. 
V= 1 —^y(l — cos 3 a) 
= 1 — \y sin 3 a 
2 p 2 
