398 



Prof. Gr. G-. Stokes on the Foci of Lines [June 2 1 , 



the principal plane of greatest and least parameters are imaginary. If 

 we take v xy , v zy to denote the inclinations to Oy of the normals to the 

 umbilici in the planes of xy, zy, we have, by the requisite interchanges 

 of letters : — 



a\a 2 -c 2 ) c 2 (a 2 -c 2 ) 



ta » 4 ^=6^wr tan v -=w^> ; * • (1o) 



If a plate be cut perpendicular to the normal at one of these umbilici, 

 one of the polarized pencils which it transmits will give the images of 

 both systems of cross lines distinct together ; and the distinctness will 

 not be affected by rotating the plate in its own plane while the cross lines 

 are fixed. In this respect it agrees with a plate of a uniaxal crystal cut 

 in an arbitrary direction, with which it might easily be confounded. But 

 if the double refraction be strong enough to give a sensible lateral separa- 

 tion of the two oppositely polarized images, the two cases may be distin- 

 guished thereby in either of two ways : — First, if the images be compared 

 with a mark fixed to the focus of the eyepiece, and the crystal be rotated 

 in its own plane, while the object viewed through it [is fixed, in the case of 

 a uniaxal crystal the image free from astigmatism will remain fixed, while 

 any point of the other describes a small circle round its mean position, 

 whereas in a plate of a biaxal crystal cut perpendicular to the normal at 

 one of the umbilici above considered it is the reverse; the image affected 

 by astigmatism remains fixed, though its distinctness alters, while any 

 point in the other describes a small circle about its mean position. 

 Secondly, if the plane of separation of the two oppositely polarized images 

 be noticed, in a uniaxal crystal the plane of polarization of the image 

 which is free from astigmatism will be parallel to the plane of separa- 

 tion, while in a biaxal crystal cut as above supposed it will be perpendi- 

 cular to the plane of separation*. 



There are no umbilici in the circular sections of the wave-surface made 

 by the principal planes. If we equate p' given by (7) to be the radius of 

 curvature in the primary plane, we get, in fact, cos 2 + sin 2 6=0, which 

 cannot be satisfied. 



The formulae (15) give for v Iy , v zy (ray D) in aragonite 69° 26' and 

 45° 8' ; in topaz 46° 49' and 55° 27'. 



In the employment of his method 3Ir. Sorby has chiefly had in view 

 the discrimination of minerals, but it admits of one or two interesting 

 applications to optical theory. 



At the time when Eresnel invented his theory of double refraction it 



* [The first test supposes the surfaces of the plate to be pretty truly parallel, ps 

 otherwise it would produce displacement in consequence of its slightly wedge-shaped 

 form ; and Mr. Sorby thinks this requirement would prevent this test from being of 

 much use. As to the second test, it is needless to observe that in any case in which we 

 know independently which is the principal plane we need not attempt to observe the 

 lateral separation of the images. — October 1877.] 



