256 Geometrical Propositions 
that the interval between them (40),—or the retardation of the former,—may be de- 
rived very easily from the letters that designate that ray. Let SPmMpMS be any 
such ray. The sum of the distances of the point S from each of the points marked 
by the letters (PmM/pM™) that denote (39) the part of the ray contained within the 
crystal, is proportional to the interval of retardation; that interval being equal to 
oy (SP+ Sm+SM+Sp+SM). Lorn 
For if from the point , where the last internal ray M emerges from the second 
surface of the crystal, a perpendicular HJ be let fall upon OS, meeting OS in J, the 
time of describing OJ with the velocity V would (43) bey (SP "— Sm" +SM" 
—Sp'+SM"). But (41) the actual time of describing the broken path PmMpM 
e (PP” + mm" + MM" + pp" + MM"); and, on inspecting the figure, this time 
ag VxOs 
is seen to be greater than the time of describing OJ, by Toa (SP + Sm +SM+ Sp 
+SM), or by the time in which the line a5 (SP+ Sm +SM+ Sp + SM)would 
be described with the velocity V. Consequently, at the moment when the light in the 
ray SPmMp MS emerges at the point EH from the second surface of the crystal, the 
light in the imaginary uninterrupted ray OS will have passed the point J by an interval 
equal to the line just mentioned ; and as the two rays afterwards have the same velo- 
city and parallel directions, this interval is the retardation of the emergent ray. 
46. The rays emerging from the first surface after any odd number of internal re- 
flections are to be compared with the ordinarily reflected ray Os to which they are 
parallel ; the light in Os, which moves with the velocity V, being supposed to leave 
O at the moment when the refracted light enters the crystal at O. The mode of 
proceeding in this case is exactly similar to that in the last, and the interval is de- 
termined in the same way, using sin place of S; the retardation of the ray SPmMps, 
for example, of which the part PmMp is contained within the crystal, being equal 
to Ae (sP +sm+sMM+ sp).* 
47. It is remarkable that the preceding demonstration nowise depends upon the sup- 
position that the planes perpendicular to the rays P,M,p,m, are tangent planes to the 
surface of refraction at the pots P,M,p,m. If we had supposed any planes—dif- 
ferent from the plane of the figure— to pass through the points P,M,p,m, and the 
rays to coincide in direction with perpendiculars let fall from O upon these planes, 
and to have velocities inversely proportional to the lengths of the perpendiculars, the 
intervals of retardation would haye remained unchanged. Hence the retardations are 
the same as if the lines OP,OM, Op,Om, were the directions of the rays in passing 
* The change of phase, which may take place at a surface of the crystal, is not here considered as af- 
fecting the intervals. ’ 
