to the Wave Theory of Light. 53 



within the circular section of the sphere, and the minus raj is 

 of course the extraordinary one. 



61. Let PJf, preserving a constant length J, move parallel 

 to itself between the surfaces of the uniaxal sphere and spheroid, 

 so as to form a ring (50). Then supposing the spheroid, with 

 the ring-edge described on it by the point _3f, to remain fixed, 

 imagine the sphere, carrying the ring edge P along with it, to 

 move parallel to PJf, from P towards Jf, through a distance 

 equal to J, and the two ring-edges will exactly coincide. 



Hence the uniaxal ring-edge is the intersection of a sphere 

 and a spheroid, the diameter of the sphere being equal to the 

 axis of revolution of the spheroid, and the line joining their 

 centres being perpendicular to the faces of the crystal and equal 

 to the breadth / of the ring. And the projection of this inter- 

 section, on a plane perpendicular to the line joining the centres 

 of the sphere and the spheroid, is the uniaxal ring-trace. 



62. The biaxal ring-edge is (51) the intersection of two equal 

 biaxal surfaces similarly posited, the line joining their centres 

 being perpendicular to the faces of the crystal and equal to the 

 breadth of the ring. And the projection of this intersection, on 

 a plane perpendicular to the line joining the centres of the sur- 

 faces, is the biaxal ring-trace* 



* In applying the general theory (51, 52) to biaxal rings, it is necessary to 

 know the equation of a biaxal surface, which may be found in the following 

 manner : Let r, r', r", be three rectangular radii of the generating ellipsoid abc, 

 the two latter being the semiaxes of the section made by a plane passing through 

 them; so that if from the centre two distances OT, 0V, equal to /, r", be taken 

 on the direction of r, the points T and V will belong (9) to the biaxal surface ; and 

 let a plane parallel to the plane of r', r", and touching the ellipsoid, cut the direc- 

 tion of r at the distance p from the centre. Then if r make the angles a, ft, 7, 

 with the semiaxes a, b, c, we shall have, by the nature of the ellipsoid, 



1 cos 2 a cos 2 cos 2 y 

 ^ = ~tf~ ~V r ~ + ~^~ ' 



p* = # 2 cos 2 a + i 2 cos 2 b + c 2 cos* -y. 

 Now, since the sum of the squares of the reciprocals of three rectangular radii of 



