18/7.] seen through a Crystalline Plate. 



391 



Through the normal EH draw the two rectangular planes of principal 

 curvature of the surface at E, and let C, C be the centres of curvature, 

 and jo, p' the radii of curvature, on the same scale in which HE represents 

 the wave-velocity v in the direction HE. Then the rays in that plane of 

 principal curvature, the normals in which intersect in C, may be thought 

 of as diverging from C in an ordinary medium, of which the refractive 

 index is v\ If - be the thickness of the plate, the distance of C from 



the second surface will be ^r, and the product of this by v, or pr, will 



give the distance of the focus in that plane after refraction into air ; and 

 therefore the apparent index of refraction will be p~ l . Similarly p'~ l will 

 will be the apparent index in the other plane. And in order that one or 

 other of the two rectangular systems of lines may be seen distinctly at 

 the proper focus, the lines must be placed perpendicular respectively to 

 the two planes of principal curvature. 



A similar construction applies to the other pencil which the plate is 

 capable of transmitting independently, to which corresponds the other 

 sheet of the wave-surface, and which is polarized in a plane perpendicular 

 to the plane of polarization of the former pencil. In a biaxal crystal, in 

 which neither sheet of the wave-surface is a sphere, there will in general 

 be four focal distances at which lines in proper directions can be seen 

 distinctly. For either pencil the two required directions are perpendi- 

 cular to each other ; and if the plate be perpendicular to one of the prin- 

 cipal planes, or planes of optical symmetry of the crystal, the required 

 directions of the cross lines are the same for both pencils, namely, 

 parallel and perpendicular to the plane of symmetry. 



The case next in order of simplicity to that of a uniaxal crystal cut 

 parallel to the axb is that of a biaxal crystal cut in a direction perpen- 

 dicular to one of the principal axes ; but before proceeding to this it 

 may be well to complete the investigation for a uniaxal crystal, by con- 

 sidering a plate cut in any manner. 



Let 6 be the inclination of the axis of the crystal to the normal to the 

 plate ; a, c, as before, the polar and equatorial semiaxes of the spheroid. 

 We need only consider the extraordinary ray and the spheroid corre- 

 sponding to it. Let p be the radius of curvature of the elliptic section 

 made by the principal plane, p the radius of curvature of the perpendi- 

 cular section, which will be the length of the normal drawn as far as the 

 axis of revolution, p and p are to be expressed in terms of 6. We 

 have 



p- 1 = a~ 2 c-\a 2 cos 2 6 + c sm 2 6)*,. (2) 



p'-i^c-^a 2 cos 2 d + c 2 sin 2 0)* (3) 



(2) gives the apparent index as obtained by focusing on a line perpen- 

 dicular to the principal plane, and (3) as obtained by focusing on a line 

 in the principal plane. 



