254 Prof. H. F. Dawes on Image 



As in other cases considered, this distance is just the same as 

 if the light were refracted from a substance of absolute 

 refractive index « 2 /2 /n/ to one of index n 2 " 2 /^i"« 



4. Image formed by direct refraction through a series of 

 plane parallel plates of uniaxial crystal all having optic axes 

 in the common normal to the surfaces. — The extraordinary 

 -wave in the first crystal will be extraordinary in all successive 

 plates and the ordinary will remain ordinary, so that only 

 two images will be formed at each stage. 



By the argument of Section 3 the image-position at each 

 stage and therefore the final image-position will be the same 

 as if the light were refracted into and out of a thin air-film 

 of negligible thickness separating each pair of plates. By 

 (7) the image is " drawn up " toward the observer a distance 

 t{\ — %/n 2 2 ), on account of refraction through any plate of 

 thickness t and indices n x and n 2 , and this "drawing up " 

 process will be repeated for each plate of the series. Hence 

 the image will be nearer to the observer than the original 

 source by the sum of all such quantities as t(l — n 1 /n 2 2 ). 

 The case of some plates being isotropic is of course included 

 in this, the corresponding displacement terms being obtained 

 by setting n x equal to n 2 . 



5. Image-formation by " direct refraction" through a lens 

 of uniaxial crystal with optic axis lying along the geometrical 

 axis. — Recall that for refraction from a medium of index n' 

 to one of index n !f through a surface of curvature 1/r, the 

 object and image distances are related by the law 



n" n' 

 v u 



where k is the dioptric power of the surface and is equal to 

 ,(V' — ?i')/r, (9). Note that this is fundamentally a relation 

 between the curvature 1/u of the incident wave and the 

 -curvature 1/v of the refracted wave. 



In the case under consideration, the light- wave P'M from 

 a point P on the axis (fig. 3) is incident on the first 

 surface M; is refracted as a spheroidal wave R'M whose 

 centre of curvature is a certain point Q and centre of radiation 

 the centre of the spheroid R ; is incident on the second 

 surface N as a wave with centre of curvature at a point S ; 

 is finally refracted into air as a homocentric pencil of 

 centre T. T is the required image-point. 



Since the light travels along the axis of the crystal with 

 the velocity corresponding to the index nj, the powers of the 

 first and second surfaces will be h' and k", equal to (n x — l)/r' 

 and (l — ni)/r", by (9). In accordance with (3) 



RM=?i 2 2 . QM/V and RN = n 2 2 . SN/S 2 , (10). 



