'J CONICAL REFRACTION. 



the elasticity of the medium is the same in two of the three direc- 

 tions, the equation of the wave-surface is resolvable into two, which 

 represent the sphere and spheroid of the Huygenian law. The 

 two optic axes in this case coincide ; and the law of Huygens is 

 thus proved to be a case of a more general law, and shown to 

 belong to uniaxal crystals only. Finally, when the elasticity is 

 the same in all the three directions, the wave-surface becomes a 

 sphere ; and the refraction is single, and takes place according to 

 the ordinary law of the sines. This case comprises a few of the 

 crystallized, and most uncrystallized substances. 



There are two remarkable cases, however, in this elegant and 

 profound theory, which its author seems to have overlooked, if 

 not to have misapprehended. In a communication made to the 

 Academy at its last meeting, Professor Hamilton has supplied 

 these omissions in the theory of Fresnel, and has been thus led to 

 results in the highest degree novel and remarkable. 



To understand these conclusions, it may be useful to revert for 

 a moment to the original theory of Fresnel. The general form 

 of the wave-surface is determined by the equation 



(a 2 cos 2 a + tf cos 2 /3 + c 2 cos 2 -y)r 4 



_ |y (ja + c 2 ) C os 2 a + & 2 ( 2 + c 2 ) cos 2 /3 4 c 2 (a 2 + ft 2 ) cos 2 7 ] r 2 

 + a 2 b 2 c 2 = ; 



in which a, j3, 7, denote the angles made by the radius- vector with 

 the three axes, and a 2 , 6 2 , c 2 , the elasticities of the medium in these 

 directions. If now we make cos /3 = in this equation, so as to 

 obtain the section of the surface made by the plane of ars, the result 

 is reducible to the form 



(r z - V] [( 2 cos 2 a + c 2 sin 2 a) r z - V] = 0. 



So that the surface intersects the plane of xz in a circle and ellipse, 

 whose equations are 



r = ft, (a- cos 2 a + c 2 sin 2 a) r 2 = V. 



Now b, the radius of the circle, being intermediate between a 

 and c, the semiaxes of the ellipse, it is obvious that the two curves 

 must intersect in four points, or cusps, as represented in (fig. 1) ; 

 and the angle which the radius-vector OP, drawn to the cusp, 



