ON DOUBLE REFRACTION. 269 
in each principal plane of a biaxal crystal the ray polarized in that plane 
obeys the ordinary law of refraction, leayes no reasonable doubt that Fresnel’s 
construction contains the true laws of double refraction, at least in their broad 
features. But regarding this point as established, I have rather in view a 
verification of those laws which admit of being put to the test of experiment 
with extreme precision ; for such verifications might often enable the mathe- 
matician, in groping after the true theory, to discard at once, as not agreeing 
with observation, theories which might present themselves to his mind, and 
on which otherwise he might have spent much fruitless labour. 
To make my meaning clearer, I will refer to Fresnel’s construction, in 
which the laws of polarization and wave-yelocity are determined by the 
sections, by a diametral plane parallel to the wave-front, of the ellipsoid * 
Cv +b y+e27=1 : (11), 
where a, b, c denote the principal wave-velocities. The principal semiaxes 
of the section determine by their direction the normals to the two planes of 
polarization, and by their magnitude the reciprocals of the corresponding 
wave-velocities. Now a certain other physical theory which might be pro- 
posed leads to a construction differing from Fresnel’s only in this, that the 
planes of polarization and wave-velocities are determined by the section, by 
a diametral plane parallel to the wave-front, of the ellipsoid 
yf we 
a Z 
gtpteele ee ee et es (12), 
the principal semiaxes of the section determining by their direction the 
normals to: the two planes of polarization, and by their magnitudes the 
corresponding wave-velocities. The law that the planes of polarization of 
the two waves propagated in a given direction bisect respectively the two 
supplemental dihedral angles made by planes passing through the wave- 
normal and the two optic axes, remains the same as before, but the posi- 
tions of the optic axes themselves, as determined by the principal indices 
of refraction, are somewhat different; the difference, however, is but small 
if the differences between a’, b,c? are a good deal smaller than the quantities 
themselves, Each principal section of the wave surface, instead of being a 
circle and an ellipse, is a circle and an oval, to which an ellipse is a near 
approximation t. The difference between the inclinations of the optic axes, 
and between the amounts of extraordinary refraction in the principal planes, 
on the two theories, though small, are quite sensible in observation, but only 
on condition that the observations are made with great precision. We see 
from this example of what great advantage for the advancement of theory 
obseryations of this character may be. 
One law which admits of receiving, and which has received, this searching 
comparison with observation, is that according to which, in each principal 
plane of a biaxal erystal, the ray which is polarized in that plane obeys the 
ordinary law of refraction, and accordingly in a uniaxal erystal, in which 
every plane parallel to the axis is a principal plane, the so-called ordinary 
ray follows rigorously the law of ordinary refraction. This law was carefully 
verified by Fresnel himself in the case of topaz, by the method of cutting 
plates parallel to the same principal axis, or axis of elasticity, carefully 
* Tt would seem to be just as well to omit the surface of elasticity altogether, and refer 
the construction directly to the ellipsoid (11). 
+ The equation of the surface of wave-slowness in this and similar cases may be readily 
obtained by the method given by Professor Haughton in a paper “ On the Equilibrium 
op _o of Solid and Fluid Bodies,” Transactions of the Royal Irish Academy, vol. xxi. 
p. 172. 
