150 Mr. Luoyp on Conical Refraction. 
which the normals to the circle and ellipse in the plane of wz make with one 
another is 
Va—b VB 
ac 2 
and it can be easily shown that the tangent of the angle which the optic axis makes 
with the normal to the circle, or the cusp-ray, is 
VE=B Ee 
nace 
Now, this is about half the former, since b?=ac, nearly ; and consequently the optic 
axis nearly bisects the angle contained by the extreme normals in the plane of zz. 
Hence if 4 and B be the intersections of the two optic axes with the sphere whose 
centre is at the cusp, and JV the intersection of one of the normals at that point 
with the same (fig. 2), the angle 4 C ranges through every magnitude between 
O and 360°, the arch V A being all the time very small. Let the angle NA C be 
denoted by a, and NP C by w, NP being the arch bisecting the angle V; then 
in the triangle 4 P N, we have 
cos. w=cos. 4 N. sin. a.sin. } N+cos. a.cos.} NV; 
or, since 4 N is very small, and therefore cos. 4 N=1, nearly, 
cos. w=cos, (a—4$ NV), and w=a—} N, nearly. 
But, when any side of a spherical triangle is very small in comparison with the other 
two, the adjacent angles are together equal to 180° q. p. Consequently, 
N=a, and w=4a, nearly. 
From this it appears that the angle which the plane of polarization of any ray makes 
with the plane of the optic axes, is half the angle which the plane passing through the 
normal and the near axis makes with the same plane. But this latter angle, it may be 
easily shown, is very nearly the same as that which the plane passing through the 
emergent ray and the axis of the cone makes with the plane of the optic axes. Con- 
sequently, the angle which the plane of polarization of any ray of the emergent cone 
makes with the plane of the optic axes is half of that which the plane containing that 
ray and the axis of the cone forms with the same plane. 
The general phenomena being observed, it remained to examine the magnitude and 
position of the emergent cone, and to compare the results with those furnished by 
theory. For this purpose I viewed the aperture in the second plate through a small 
telescope, which was moved ina plane nearly perpendicular to the axis of the emer- 
gent cone ; and by noting the points at which the light failed, I obtained the magni- 
tude of the section of the cone made by that plane. ‘Lhe distance of this section from 
