Rev. H. Lloyd on the Phenomena of Light. 113 



examine for a moment the form of the wave. Its equation, 

 referred to polar coordinates, is 



(a 4 cos 2 a + b 1 cos 2 B -f c* cos 2 y) r* 

 - [a 2 (b 2 -f c 2 ) cos 2 a + b* (a 2 + c 2 ) cos 2 B 4- c 2 {a 2 + 6 2 ) cos* y] r* 

 + a 2 & 2 C* = 

 in which a, B, y, denote the angles made by the radius vector 

 with the three axes of coordinates. If now we make cos y = 

 in this equation, so as to obtain the section of the surface 

 made by the plane of xy, the result is reducible to the form, 



(r 2 —c 2 ) [(a 2 cos* u + b 2 sin 2 a) r 2 — a 2 6 2 ] = 0; 



so that the surface of the wave intersects the plane of xy in a 

 circle and ellipse, whose equations are 



r = c, (a 2 cos 2 a + b 2 sin* a) r 2 = a 2 5f. 

 Now if c, the radius of the circle, be intermediate between 

 a and b, the semiaxes of the ellipse, the two curves will inter- 

 sect in four points, or cusps; and the angle which the radius 

 vector drawn to the cusp makes with the axis of a, is found 

 by eliminating r between the two equations, by which means 

 we obtain a hzZfrT 



sin a 



- c V a" 



b*' 



At each of the points thus determined, there will be two 

 tangents to the plane section, and therefore two tangent planes 

 to the surface; and consequently a single ray, proceeding 

 within the crystal to one of these points, will at emergence be 

 divided into two, whose directions are determined by those of 

 the tangent planes. 



Such seems to have been Fresnel's conception of this case. 

 Professor Hamilton has shown, however, that there is a cusp 

 at each of these points, not only in this particular section, 

 but in every section of the wave-surface passing through the 

 line whose direction has just been determined ; or that there 

 are, in fact, four conoidal cusps on the general wave-surface at 

 the 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, proceeding 

 from a point within the crystal in any of the above-mentioned 

 directions, ought to be divided into an infinite number of 

 emergent rays, forming a cone of the 4th order. 



It is evident, further, that the circle and ellipse which thus 

 intersect must have four common tangents. Fresnel has shown 

 that the planes passing through these tangents, and parallel 

 to the 3rd or mean axis, are parallel to the circular sections 

 of a curved surface which he calls the surface of elasticity; 

 and he seems to have concluded that these planes touched 



Third Series. Vol. 2. No. 8. Feb. 1833. Q 



