1877.] 



seen through a Crystalline Plate. 



389 



ordinary pencil, which is polarized in the axial plane, at which both 

 systems at once will be seen distinctly. And the ordinary index, which 

 will be given by the ordinary image, will be a geometric mean between 

 the two apparent extraordinary indices, of which one, namely, that got 

 from the lines in the axial plane, will be the real extraordinary index. 



There are two imiaxal crystals, calcite and quartz, for which we know 

 accurately the principal refractive indices for the principal lines of the 

 spectrum from the measures of Hudberg. The principal indices for these 

 two minerals and the apparent indices in the two directions mentioned 

 above are given in the following Table. The indices are given to four 

 places of decimals, and the fixed lines C, D, E are chosen, whence the 

 results applicable to the kinds of light most likely to be employed may 

 be obtained, directly or by interpolation. 



Lines. 



Calcite. 



1 



uartz. 











% 



% 





a! 













w 













C ... 



1-6545 



1-4846 



1-3321 



1-8438 



1-5418 



1-5509 



1-5601 



1-5328 



D ... 



1-6585 



1-4864 



1-3322 



1-8505 



1-5442 



1-5533 



1-5624 



1-5352 



E ... 



1-6636 



1-4887 



1-3322 



1-8590 



1-5471 



1-5563 



1-5656 



1-5380 



It is well known that the double refraction of quartz differs from that 

 of the generality of uniaxal crystals. Its wave-surface for any colour, 

 instead of being the sphere and spheroid of Huyghens, is a surface of two 

 distinct sheets, which, instead of touching, only make a very close 

 approach along the axis. The polar diameters of the outer, or ordinary, 

 and of the inner, or extraordinary, sheet differ by minute and practically 

 equal quantities from the equatorial diameter of the ordinary sheet. 

 The effect of this, however, on the indices, real or apparent, determined 

 by Mr. Sorby's method on a plate cut perpendicular to the axis, w T ould 

 not be sensible. The peculiarity w T ould show itself by giving the two 

 images at different depths circularly polarized, one right-handedly and the 

 other left-handedty. 



It may be noticed that the refractive index is given by the reciprocal 

 of the radius of curvature of a section of the wave-surface by a plane 

 perpendicular to the lines seen in focus, and that in order that the lines 

 may be seen distinctly, they must be perpendicular to one of the planes 

 of principal curvature. This rule, as I proceed to show, is general ; and 

 it will much simplify the calculation in more complicated cases, by 

 enabling us to dispense with the direct application of Huvghens's con- 

 struction. 



Let O be a point in the first surface of the plate, and consider a small 



