Mr. Luioyp on Conical fefraction. 151 
the crystal being then measured, the angle of the cone was obtained from the trigo- 
nometrical tables ; and was found to be very nearly 6°. I then placed the flame of a 
wax taper at the centre of the section, and removing the plate from the second surface 
of the crystal, found the direction of the ray reflected from the surface. A well 
defined mark was then placed on this line, at a considerable distance, and the angular 
distance between the centre of the flame and the mark measured by a sextant, whose 
centre was brought exactly to the place of the crystal. This angle was found to be 
31° 56;; and consequently the angle of emergence corresponding to the central rays 
of the cone was 15° 58’. 
Now to compare these results with those of theory.—lIt is a well-known principle 
of the theory of waves, that the direction of a ray incident upon, or emergent from, 
a crystal, and the normal to the front of the wave, are always in the same plane per- 
pendicular to the surface of incidence or emergence ; and the angles which these two 
lines make with the perpendicular to the surface, are connected by the known law of 
the sines; the index of refraction being the reciprocal of the normal velocity of the 
wave, or of the perpendicular upon the tangent plane. Now, at the cusp, there are 
an infinite number of normals to the wave, and consequently an infinite number of 
corresponding emergent rays. Of these the two rays in the plane of the optic axes 
form the greatest angle, and their directions are determined by those of the normals 
to the circle and ellipse, which constitute the section of the wave-surface in that plane. 
If then » and p’ denote the angles of emergence of these rays, «the angle which 
the normal to the circle, or cusp-ray, makes with the perpendicular to the surface, 
athe angle contained by the normals to the circle and ellipse, and p the perpendi- 
cular from the centre on the tangent to the ellipse at the cusp, we have 
c ae : bars 
sin. p = > sin. ct, sin. p! = ji sin. («—a) ; 
é 
Tn which 
1 2 0262" ¢— 8? ./82—e2 
Il Vatt+e Li Wg ia b b?—e 
Pp ac ac 
Now in Arragonite, according to the determination of M. Rupsere, 
1 
'_ 1.5326, } = 1.6863, - = 1.6908 ; 
a c 
And substituting these values we find 
1 
i = 1,68708, a = 1°.44).48". 
These values being introduced in the first two equations, ’ and pe" will be deter- 
mined for any given surface of emergence. In this manner Professor HamiLron has 
found that when «=O, or the surface of.emergence perpendicular to the cusp-ray, 
pe! =0, and p'= 2°.56'-51". And when. = 9°. 56'-27", or the surface perpendicular 
