CONICAL REFRACTION. 



3 



makes with the axis of x, is found by eliminating r between the 

 two equations, by wliich means we obtain 



Fig. 1. 



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

 to tho plane section ; and consequently the ray OP, proceeding 

 within the crystal to one of these points, 

 might be supposed to be divided at emer- 

 gence into two, whose directions are deter- 

 mined by those of the tangents. 



Such seems to have been Fresnel's con- 

 ception 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 OP ; or, in 

 other words, that there is a conoidal cusp on 

 the general wave-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 rat/, such as OP, pro- 

 ceeding from a point within the crystal to one of these points, 

 must be divided into an infinite number of emergent rays, constitut- 

 ing 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 elastic it;/ of 

 Fresnel's theory, or perpendicular to the optic axes. Fresnel seems 

 to have concluded that these planes touched the wave-surface only 

 in the two points just mentioned ; and, consequently, that a single 

 ray, incident upon a biaxal crystal in such a manner that one of 

 the refracted rays should coincide with an optic axis, would be 

 divided into two, determined by the points of contact. This result, 

 if verified by experience, would place a remarkable distinction 

 between the phenomena of uniaxal and biaxal crystals ; but though 

 the case was examined by M. Biot, no corresponding appearances 

 were observed. 



