146 Mr. Luioyp on Conical Refraction. 
To understand these conclusions, it may be useful to revert for a moment to the 
original theory of Fresnet. The general form of the wave-surface is determined by 
the equation 
(a cos.'a + b° cos." + c*cos.* y) r* 
—[a'(B + c*)cos.°a + b*(a* + c*) cos.7B + ¢°(a® + b*)cos.*y] r° 
+ab'?=0; 
in which a, 3, y, denote the angles made by the radius-vector with the three axes, and 
a’, 6°, c’, the elasticities of the medium in these directions. If now we make cos. 8 =0, 
in this equation, so as to obtain the section of the surface made by the plane of xz, the 
result is reducible to the form 
(7° — b?) [(a@’cos.’a + e’sina) 7?—a’c?] =0. 
So that the surface intersects the plane of xz in a circle and ellipse, whose equations 
are 
r=b, (a’cos”a+e'sin.” a) =a’ Cc’. 
Now 4, the radius of the circle, being intermediate between a and c, the semiaxes of 
the ellipse, it is obvious that the two curves must intersect in four points, or cusps, as 
represented in (fig. 1); and the angle which the radius-vector OP, drawn to the cusp, 
makes with the axis of z, is found by eliminating r between the two equations, by 
which means we obtain 
s a Bc? 
sin.a= 2 Of Tere 
At each of the points thus determined there will be two tangents to the plane section ; 
and consequently the ray OP, proceeding within the crystal to one of these points, 
might be supposed to be divided at emergence into two, whose directions are deter- 
mined by those of the tangents. 
Such seems to have been Fresnet’s conception of this case. Professor HamiLTon 
has shown, however, that there is a cusp at each of these points, not only in this parti- 
cular section, but in every section of the wave surface passing through the line OP ; 
or, in other words, that there is a conoidal cusp on the general waye-surface at the 
four points of intersection of the circle and ellipse ; so that there must be an infinite 
number of tangent planes at each of these points, and consequently a single ray, such 
as O P, proceeding from a point within the crystal to one of these points, must be divided 
into an infinite number of emergent rays, constituting a conical surface. 
It is evident further, that the circle and ellipse will have four common tangents, 
such as MN (fig. 1.) The planes passing through these tangents, and parallel to the 
third or mean axis, are parallel to the circular sections of the surface of elasticity of 
Fresnex’s theory, or perpendicular to the optic axes. FResneL seems to have con- 
