136 Professor Hamitton’s Third Supplement 
For we hava’@en that at a conoidal cusp the tangent plane to the wave is indeter- 
minate ; it is evident therefore that a cusp-ray must correspond to an infinite variety 
of systems of ponents of normal slowness o, 7, v, within the biaxal crystal, and 
therefore also to an infinite variety of systems of direction-cosines a, 8, y, of the 
external ray ; so that this one internal cusp-ray must correspond to an external cone 
of rays, according to a new theoretical law of light, which may be called ExtTerna 
ContcaL REFRActION. 
And again, at a circle of contact, the wave has one common tangent plane for all 
“the points of that circle, and therefore the infinite variety of internal rays which cor- 
respond to these different points have all one common wave-normal, which may be 
called a normal of circular contact, and all these internal rays have one common 
system of components of normal slowness o 7 » within the crystal, and consequently 
correspond to one common external ray : so that this one external ray is connected 
with an internal cone of rays, according to another new theoretical law of light, 
which may be called Inrernat Contcau Rerraction. 
To develope, somewhat more fully, these two new consequences from FRESNEL’s prin- 
ciples, let us begin by considering external conical refraction : and let us seek the 
equation of the external cone of rays, corresponding to the internal cusp-ray p’. The 
approximate equation (G") of the wave, near the end of this cusp-ray, in the trans- 
formed co-ordinates x 7, 2,, gives the following approximate expression for the undu- 
latory slowness v of a near ray, considered as a homogeneous function of the first 
dimension of the cosines a, 3, y, of its inclinations to the positive semiaxes of these 
co-ordinates 2, ¥, Z,, 
4 =b- iets (a,+ Va? + 2?) Ci?) 
in which 
=4b Vo_—b? Vb?—a; 2 @z) 
it gives therefore by our general method, the following components of normal slow- 
ness parallel to the same semiaxes of wy, 2,, 
i —¢@= oe tacts Be 
gi aL oa eke PES) 
m= = we =hee OB ie A) ise (v") 
op, Vaz+Be { 
2 , ov 
# il hace Papua 
the expressions for «, r, becoming itidefinitely more accurate as a, 8, diminish, that is, 
as the near internal ray approaches to the cusp-ray ,, and the expression for v’ being 
