to the Wave Theory of Light. 41 



therefore 



OPS POL = PP" 

 cos POP, ~ OL ' 



Hence the projection is equal to 9 



t 



If the path of a ray P be projected on the incident ray OS, 

 then producing OS to meet PP / in /, we see, by what has just 

 been proved, that the length of the projection is equal to 



PP" SP" 



) ~oT 9 os~' 



by similar triangles. In like manner, the projection's of the 

 paths of rays M, p, m, on the direction of the incident ray OS, 

 are equal to 



SM" Sp" Sm" 



~> " 



OS OS' OS ' 



respectively. 



43. Let each rectilinear path be measured in the direction 

 in which the light moves along it ; and according as the 

 - direction so measured makes an acute or an obtuse angle with 

 the direction OS, measured from to S, let the projection of 

 the path on OS be reckoned positive or negative. Then if 

 SPmMpMS be any ray entering the crystal at 0, and emerging 

 from its second surface at E, and if a perpendicular El be let 

 fall from E upon OS, meeting OS in /, the distance 01, from 

 to the foot of this perpendicular, will evidently be equal to 

 the algebraic sum of the projections of the paths P, m, M, p, M, 

 contained within the crystal, taking each projection with its 

 proper sign. It is obvious that the projections of the P and M 

 rays are always positive. And as the lines Op', Om' the 

 directions of the rays p, m lie in planes which are respectively 

 perpendicular to pp fl mm,, or to Op", Om", it is easy to see that 

 these directions make acute or obtuse angles with OS, according 

 as the points p", m", lie below the point S or above it ; that is, 

 the projections are positive or negative according as the points 

 p", m", lie without the circle OS towards P, M, or within the 



