262 Geometrical Propositions 
If OP" be perpendicular to a tangent plane at P, the vibrations of the ray P will 
be perpendicular to @P-and will-ie-in the plane POP’. A similar remark applies 
to the rays M, p, m. 
60. When two semiaxes a,b, of the ellipsoid abc become equal, it changes into a 
spheroid aac described by the revolution of the ellipse ac about the semiaxis ¢ ; and 
the biaxal aac, generated by this spheroid, is * composed of a sphere whose radius is a, 
and a concentric spheroid acc described by the revolution of the ellipse ac about the 
semiaxis a; so that, the diameter of the sphere being equal to the axis of revolution 
of the spheroid, the two surfaces touch at the extremities of the axis. This combina- 
tion of a sphere anda spheroid is the surface of refraction for uniaxal crystals. Jn 
these crystals, therefore, the refracted ray whose direction is determined by the inter- 
section of the right line #S with the surface of the sphere follows the ordinary law 
of a constant ratio of the sines, and is called the ordinary ray ; whilst the other, 
whose variable refraction is regulated by the intersection of FS with the spheroid, is 
ealled the extraordinary ray: And hence uniaxal crystals are usually divided into the 
two classes of positive and negative, according to the character of the extraordinary 
ray ; being called positive when it is the plus ray, and negative when it is the minus 
ray. The first case evidently happens when the spheroid is oblate, and therefore lies 
without the sphere described on its axis ; the second, when the spheroid is prolate, 
and therefore lies within the sphere. The second case, (which is that of Iceland 
spar,) may be supposed to be represented in the figure (Fig. 8), where the elliptic 
section of the spheroid, made by a plane of incidence oblique to the axis, lies within 
the circular section of the sphere, and the minus ray is of course the extraordinary 
one. 
61. Let PM, preserving a constant length J, move parallel to itself between the 
surfaces of the uniaxal sphere and spheroid, so as to form a ring (50). Then sup- 
posing the spheroid, with the ring-edge described on it by. the point JZ, to remain 
fixed, imagine the sphere, carrying the ring-edge P along with it, to move parallel to 
PM, from P towards M, through a distance equal to J, and the two ring-edges will 
exactly coincide. iy 
Hence the uniaxal riag-edge is the intersection of a sphere anda spheroid, the 
diameter of the sphere being equal to the axis of revolution of the spheroid, and the 
line joining their centres being perpendicular to the faces of the crystal and equal to 
the breadth J of the ring. And the projection of this intersection, on a plane perpen- 
dicular to the line joining the centres of the sphere and the spheroid, is the uniaxal 
ring-trace. 
62. The biavxal ring-edge is (51) the intersection of two equal biaxal surfaces 
similarly posited, the line joining their centres being perpendicular to the faces of the 
* Trans. R. I. A. Vol. XVI. p. 77. 
