4 4 Geometrical Propositions applied 



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

 in 06-, which moves with the velocity F, being supposed to leave 

 O at the moment when the refracted light enters the crystal at 

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

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

 using s in place of S ; the retardation of the ray SPmMps, for 

 example, of which the part PmMp is contained within the 

 crystal, being equal to 







-=-=; (sP + sin + sM + sp)* 

 C/o 



47. It is remarkable that the preceding demonstration in no- 

 wise depends upon the supposition that the planes perpendicular 

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

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

 planes different from the plane of the figure to pass through 

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

 with perpendiculars let fall from upon these planes, and to 

 have velocities inversely proportional to the lengths of the 

 perpendiculars, the intervals of retardation would have remained 

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

 OP, OM, Op, Om, were the directions of the rays in passing 

 through the crystal, as will appear by conceiving the planes 

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



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



OP 



with an index equal to -y^-, it would take the direction OP ; if 



OM 



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



0o 



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

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

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

 said, that they would be in complete accordance, respectively, 

 with the rays SPS and SMS. 



* The change of phase, which may take place at a surface of the crystal, is not 

 here considered as affecting the intervals. 



