71 Mr. Sylvester's Analytical Development 



/ (W "" ^\ a // fc2 - cs \ 

 sm . = ^ I -,— P I = - y ( -^— ^) 



These lines I shall call by way of distinction the prime radii*. 

 Cor. (1.) If r, 7-2 be the two values of r corresponding to 

 the same values of ^^ »y, we have 



sm (, . sm I. 



which proves the celebrated problem of two rays having a 

 common direction in a crystal. 



Cor. (2.) The intersection of any concentric sphere with 

 the wave surface is formed by making r constant. Hence 

 */ i *// becomes constant, and .*. ^ ij :t '^ ^ii — constant. Hence 

 the curve of intersection is the locus of points, the sum or dif- 

 ference of whose distances from two poles when measured 

 by the arcs of great circles is constant ; the poles being the 

 points in which the prime radii pierce the sphere. 



In three cases these spherico-ellipses or spherico-hyperbolse 

 become great circles: 



1°. When »^ -j- * = the angle between the two poles, in 

 which case the curve of intersection is the great circle which 

 comprises the two poles. 



2^. When »^ — i = when the locus is a great circle per- 

 pendicular to the former and bisecting the angle between the 

 optic axes. 



3°. When i, + * = 180 when the locus is a great circle 

 perpendicular to the two above, and bisecting the supple- 

 mental angle between the two axes. 



Various other properties may be with the greatest simplicity 

 deduced from the radio-angular equation. The hurry of the 

 press leaves me time only to subjoin the following 



Proposition. 

 "To find the inclination of the radius vector to the tangent 

 plane, in terms of the angles which the radius vector makes 

 with the prime radii." 



♦ Upon the authority of Professor Airy I have appropriated the term 

 optic axee to the linei normal to the fronts of single velocity. 



