applied to the Wave Theory of Light. 263 
erystal and equal to the breadth of the ring. And the projection of this intersec- 
tion, on a plane perpendicular to the line joining the centres of the surfaces, is the 
biawal 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 7, 7’, 7’, be three rectangular radii of 
the generating ellipsoid abe, the two latter being the semiaxes of the section made by a plane passing 
through them ; so that if from the centre O two distances OT, OV, equal to 7’, 7’, be taken on the di- 
rection of 7, thepoints Tand V will belong (9) to the biaxal surface ; and let a plane parallel to the plane 
of 7, x”, and touching the ellipsoid, cut the direction of r at the distance p from the centre. Then if 7 
make the angles a, 8, 7, with the semiaxes a, b, c, we shall have, by the nature of the ellipsoid. 
1 costa cos’ costy 
a a oe as 
> 
» po=atcos?a+l*cos*h + c*costy. 
Now since the sum of the squares of the reciprocals of three rectangular radii of an ellipsoid is con. 
stant, as well as the parallelopiped described on three conjugate semidiameters, we haye the equations 
1 1 1 1 1 1 
a ae 
pT tea c; 
Or, 
1 1 1 1 1 cos*a cos? cos* 
a he = +5 ( + Bea ") = M, 
we At” Bt” Ge 
2 2 
a &? ce 
1 a®cos?a + 5°cos*B + c*cos?y 
= =N- 
rere ace 
Whence it appears that 7’, rv’, are the yalues of p in the equation 
-——4+ = 0, 
in which p denotes indifferently either semidiameter, OT or OV, of the biaxal surface. Therefore 
putting for Wand J their values, and writing z ¥ 2, instead of cos a, cos PB, cos y, and 22+y*+2° 
instead of p?, we obtain, for the equation of the biaxal surface, 
(a* +y° +2°)(a%a* +b? +e°2z*) + a2(b?+¢°)a*~— h(a +e*)y?—e(a? +62) z* + a*b%e° =0. 
‘This is the equation of the surface of refraction for a biaxal crystal in which a, 6, ec, are (54) the three 
principal indices of refraction, taking O.S' the radius of the sphere to be unity. The left-hand member 
of the equation is therefore the expression supplied by the theory of FRESNEL for the function UJ in art. 
51. : 
When the faces of the crystal are parallel to any of the principal planes of the ellipsoid, —to the plane 
of ay for example,—the nature of the ring-trace may be found very easily. For if the difference of the 
two values of z, deduced from the preceding equation of the surface of refraction, be put equal to a con- 
“stant quantity JZ, the result, when cleared of radicals, will be an equation of the fourth degree in x and y, 
which will be the equation of the corresponding ring-trace. This is a case that occurs frequently in prac- 
tice; the crystal being often cut with its faces perpendicular to the axis of x or of z, because these lines 
ect the angles made by the optic axes. : 
VOL. XVII. ; 3H 
