CONICAL REFRACTION. 9 



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 ex- 

 amine 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 in a plane nearly perpendicular to the 

 axis of the emergent cone ; and by noting the points at which the 

 light failed, I obtained the magnitude of the section of the cone 

 made by that plane. The distance of this section from the crystal 

 being then measured, the angle of the cone was obtained from the 

 trigonometrical 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 dis- 

 tance, 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. It 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 perpen- 

 dicular 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 refrac- 

 tion 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 conse- 

 quently 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 p' and /" denote the angles of 

 emergence of these rays, t the angle which the normal to the 

 ircle, or cusp-ray, makes with the perpendicular to the surface, 

 a the angle contained by the normals to the circle and ellipse, 



