82 Mr. Sylvester on the Optical Theory of Crystals, 



,'. the projection of the locus of r upon a plane drawn at N perpen- 

 dicular to the line joining N with the centre O is given by the =" 



p = ON.cotND.cos.6, 



N being the origin and the projection of N D the prime radius ; 

 which is the = " to a circle passing through N, and whose diameter 

 -=ON.cot.ND. 



Hence at the extremity of each prime perpendicular the tangent 

 plane meets the surface in a circle passing through that extremity and 

 whose radius = ^ 6 . cot a, a being to be found from the equation 



sin (2 E + g) « sin (E + e) 

 sin a sin ( E — e) 



i. e. tan .(£ + «) = (tan E)- . cot e 



Just in the same way it may be shown that the traces of the per- 

 pendiculars to the tangent planes of the surface at the point where 

 it is pierced by any prime radius upon a plane perpendicular to that 

 radius at its extremity, is also a circle passing through it, and curved 

 in an opposite direction from the circle of plane contact nearest to it. 



Hence the enveloping cone at these points may be described as 

 being perpendicular to the circular cone, formed by drawing lines 

 from the centre to the above described circle ; i.e. every generating 

 line of the one will be perpendicular to the generating line which it 

 meets of the other. 



More generally it easily appears from the last figure but one that 

 if a series of great circles (representing meridian planes) be taken 

 intersecting the great circle N R R' N' in a] fixed point, a plane 

 perpendicular to the radius passing through that point, will intersect 

 the cone of rays as well as the cone of perpendiculars corresponding 

 to those meridian planes, in two circles. So that there exist an in- 

 definite number of circular cones of rays corresponding to circular 

 cones of perpendiculars touching each other in a line lying in the 

 plane containing the extreme axes, and having their circular sec- 

 tions perpendicular to that line. 



The cusps are explained by the cone of rays degenerating into 

 a right line, and the circles of plane contact by the cone of perpen- 

 dicular so degenerating. 



Furthermore I observe in conclusion that when a ray is given it 

 follows from the general geometrical construction above that there 

 will be two meridian planes according as we take R with R', or with 

 a point 180 degrees from R', and consequently these two planes will 

 be perpendicular to each other. 



And similarly when a normal is given there will be two meridian 

 planes perpendicular to each other. 



Thus the planes passing through any radius and the two normals 

 at the points where it pierces the wave surface, are perpendicular 

 to each other, as are ako the two planes passing through any nor- 

 mal and its two corresponding radii. 



Moreover a glan ce at the figure in vol. xi. p. 541 ., will show that the two 

 lines of vibration corresponding to any front, lie respectively in the 



