42 Geometrical Propositions applied 



circle. Therefore the distance 07, in the case of the figure, i 

 equal to 



^ (SP" - 8m" + 811" - Sp" + SM") . 



44. If the paths of rays P, M 9 p, m, be projected on th 

 direction Os of the ordinarily reflected ray, the lengths of thei 

 projections will be 



sP" sM" sp" sm" 



"OS' "OS' "OS' ~Q8' 



respectively. The projections upon Os of the raysj*?, m, will b< 

 always positive ; and the projections of the rays P, J/, will b< 

 positive or negative according as the points P", Jf", lie abov< 

 the point s or below it ; that is, according as the points P", M" 

 lie without the circle OS towards p and m, or within the circle 

 So that if SPmMps be a ray entering the crystal at anc 

 emerging from the first surface at e, and if a perpendiculai 

 ei be let fall from e upon Os, the distance Oi, from the point C 

 to the foot of this perpendicular, or the algebraic sum of the 

 projections of the paths P, m, M, p, contained within the 

 crystal, will be equal to 



in the case of the figure. 



45. Let us imagine that the light in the incident ray S'O, 

 instead of being interrupted at by the crystal, had continued 

 to move with the same velocity V in the same right line OS, 

 leaving the point at the moment when the refracted light 

 enters the crystal at 0. Comparing the light in this imaginary 

 ray with that in a ray emerging parallel to it from the second 

 surface of the crystal, after an even number of internal reflec- 

 tions, we shall find that the emergent is behind the imaginary 

 ray, and that the interval between them (40) or the retar- 

 dation of the former may be derived very easily from the 

 letters that designate that ray. Let SPmMpMS be any such 



