5 2 Geometrical Propositions applied 



passing through the nodal diameters of the surface of refraction ; 

 the bisected angle being that which contains the least semiaxis 

 c of the generating ellipsoid. The plane of polarization of the 

 ray p is found in like manner. But for the rays M, m, the 

 angle to be bisected is that which contains within it the greatest 

 semiaxis a. 



If OP f be perpendicular to a tangent plane at P, the vibra- 

 tions of the ray P will be perpendicular to OP, and will lie in 

 the plane POP'. A similar remark applies to the rays M,p, m. 



60. When two semiaxes #, b, of the ellipsoid abc become 

 equal, it changes into a spheroid aac described by the revolution 

 of the ellipse ac about the semiaxis c ; and the biaxal aac, gene- 

 rated by this spheroid, is* composed of a sphere whose radius is a, 

 and a concentric spheroid ace described by the revolution of the 

 ellipse ac about the semiaxis a ; so that, the diameter of the 

 sphere being equal to the axis of revolution of the spheroid, the 

 two surfaces touch at the extremities of the axis. This combina- 

 tion of a sphere and a spheroid is the surface of refraction for 

 uniaxal crystals. In these crystals, therefore, the refracted ray 

 whose direction is determined by the intersection of the right line 

 MS with the surface of the sphere follows the ordinary law of a 

 constant ratio of the sines, and is called the ordinary ray ; whilst 

 the other, whose variable refraction is regulated by the intersec- 

 tion of JRS with the spheroid, is called the extraordinary ray. And 

 hence uniaxal crystals are usually divided into the two classes 

 of positive and negative, according to the character of the extra- 

 ordinary ray ; being called positive when it is the plus ray, and 

 negative when it is the minus ray. The first case evidently 

 happens when the spheroid is oblate, and therefore lies without 

 the sphere described on its axis ; the second, when the spheroid 

 is prolate, and therefore lies within the sphere. The second case 

 (which is that of Iceland spar) may be supposed to be repre- 

 sented in the figure (Fig. 15), where the elliptic section of the 

 spheroid, made by a plane of incidence oblique to the axis, lies 



* Transactions of the Royal Irish Academy, VOL. xvr., p. 77 (supra, p. 12). 



