44 Geometrical Propositions applied 



ordinarily reflected ray Os to which they are parallel, the lig 

 in Os, which moves with the velocity F", being supposed to lea 

 at the moment when the refracted light enters the crystal 

 0. The mode of proceeding in this case is exactly similar 

 that in the last, and the interval is determined in the same WE 

 using s in place of 8 ; the retardation of the ray SPmMps, j 

 example, of which the part PmMp is contained within t 

 crystal, being equal to 



r\ 



(sP + sm + sM + sp)* 



OS 



47. It is remarkable that the preceding demonstration in r 

 wise depends upon the supposition that the planes perpendicul 

 to the rays P, M, p, m, are tangent planes to the surface 

 refraction at the points P, M, p, m. If we had supposed a: 

 planes different from the plane of the figure to pass throui 

 the points P, M, p, m, and the rays to coincide in the directi 

 with perpendiculars let fall from upon these planes, and 

 have velocities inversely proportional to the lengths of t 

 perpendiculars, the intervals of retardation would have remain 

 unchanged. Hence the retardations are the same as if the lir 

 OP, OM, Op, Om, were the directions of the rays in passi: 

 through the crystal, as will appear by conceiving the plar 

 that we have spoken of to be perpendicular to these lines. 



If the incident ray S'O were refracted in the ordinary w 



OP 



with an index equal to 7^, it would take the direction OP : 



(/o 



it were refracted, in like manner, with the index -^, it won 



Oo 



take the direction OM ; and if the two rays, thus ordinari 

 refracted, were to emerge from the second surface of the crysl 

 in directions parallel to OS, it is evident, from what has be 

 said, that they would be in complete accordance, respective] 

 with the rays SPS and SMS. 



* The change of phase, which may take place at a surface of the crystal, is i 

 here considered as affecting the intervals. 



