1877.] 



seen through a Crystalline Plate. 



387 



The simplest case is that of a uniaxal crystal, such as Iceland spar, cut 

 perpendicular to its axis. As regards the ordinary ray, a plate cut from 

 a uniaxal crystal, in whatever direction, behaves, of course, like a plate 

 of glass, so far as focusing is concerned, and the index obtained is the 

 true ordinary index. To find what takes place as regards the extraor- 

 dinary ray, we must have recourse to Huyghens's construction. 



Let be any point in the further surface of the crystalline plate, OA 

 perpendicular to the surface the direction of the axis, OP the direction 

 of any extraordinary ray. Let the plane of the paper be the plane of 



Fig. 1. 









D 



incidence, AOP ; take OA to represent the velocity of propagation {a) 

 within the crystal in the direction of the axis, and OD in OA produced 

 to represent the velocity of propagation (unity) in air. With as 

 centre, construct the half-spheroid, BAG, which is the extraordinary sheet 

 of the wave-surface, and the hemisphere EDF representing the wave 

 into which a disturbance emanating from would have spread in air in 

 a unit of time, and let OB or OC be denoted by c. Let OP cut the half- 

 spheroid in P. At P draw a tangent plane to the spheroid, the trace of 

 which on the surface of the crystal is projected in T ; and through the 

 trace T draw a tangent plane to the hemisphere, touching it in Q, and 

 join OQ. Then if an extraordinary ray travel within the crystal in the 

 direction OP, the refracted ray to which it will give rise will travel in a 

 direction parallel to OQ. Hence if we now take OP to denote the whole 

 path of the ray within the plate, and draw ~Pq parallel to QO, cutting 

 OA in q, the ray OP, after refraction at P, will proceed as if it eame 

 from q. Hence the limiting position of q, as P moves up to A, will be 

 the geometrical focus, after refraction, of a small pencil proceeding from 

 0, and having OA for its axis. 



Draw PAC perpendicular to OA, and let m represent the ratio of the 

 sine of refraction to the sine of incidence. Then 



w = OP : Tq ; 



'Annalen' (1862, vol. xxvii. p. 563), containing some elaborate observations on the 

 focal lines formed within a doubly refracting plate, as his experiments were made in a 

 very different manner from those of Mr. Sorby, and with a totally different object in 

 view.— October 1877.] 



