crystalline Reflexion and Refraction. 35 
meters of the index-surface, the formule (13) give 
% = h(a? +c’) —}(@ —c’) cos (6, + 4), (15) 
s? = (a +c’) —4(a@ — ©’) cos (0, — 4) 5 
observing that s and s’, the two normal velocities of propagation, are the recipro- 
cals of the radii of this surface which coincide with the wave-normal. Subtracting 
these expressions, we get 
s’ — s° = (a* — c’) sin 8, sin 6,. (16) 
As the position of the tangent plane, at any point T of a biaxal surface, de- 
pends on the position of the axes of the section QOy’ made in the generating 
ellipsoid by a plane perpendicular to OT, it is obvious that when this section is 
a circle, that is, when the point T is a node of the surface, the position of the 
tangent plane is indeterminate, like that of the axes of the section; and it is easy 
to show that the cone which that plane touches in all its positions is of the second 
order. Again, when the section ROy’ of the reciprocal ellipsoid is a circle, the 
right line OS is given both in position and length; and the tangent plane, 
which cuts OS in §, is fixed; but the point of contact T is not fixed, since the 
semiaxis OR, to which the right line ST is parallel, may be any radius of the 
circle ROy’. In this case, the pomt T describes a curve in the tangent plane, 
and this curve is found to be a circle. But both these cases have been fully 
discussed elsewhere.* 
SECT. V.—CONDITIONS TO BE SATISFIED WHEN LIGHT PASSES OUT OF ONE 
MEDIUM INTO ANOTHER.—REMARKABLE CIRCUMSTANCE CONNECTED WITH 
THEM.—RELATIONS AMONG THE TRANSVERSALS OF THE INCIDENT, RE- 
FLECTED, AND REFRACTED RAYS. 
Now let light pass out of one medium into another, suppose out of an ordi- 
nary into a doubly refracting medium; and taking the origin of rectangular 
coordinates .7,, yy, z, at a point O on the surface which separates the two media, 
let this surface be the plane of 2, y,. Then if the components of the displace- 
ment of a particle whose initial coordinates are 25, %) 2) be denoted by &, 1» €) 
* Ibid. vol. xvii. pp. 245, 260. 
F 2 
