applied to the FVave Theory of Light. 255 
If the path of a ray P be projected on the incident ray O'S, then producing OS to 
meet PP, in /, we see, by what has just been proved, that the length of the projec- 
Ni s 
tion is equal to 0 o =0 0) aby similar triangles. In like manner, the projections 
of the paths of rays MM, p, m, on the direction of the incident ray OS, are equal to 
ee Xe Sp" ‘s) Sm* , respectively. 
Os Os 
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 O 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 O, and emerging from its second surface at H, and if a perpendicular 
EI be let fall from LZ upon OS, meeting OS in I; the distance OF, from O to 
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 asthe lines Op’,Om’',—the directions of the rays p, m,— 
lie in planes which are respectively perpendicular to pp,, mm,, or to Op", Om", it 
is easy to see that these directions make acute or obtuse angles with O.S, according as 
the points p”, m”, lie below the point S or above it ; that is, the projections are posi- 
tive or negative according as the points p”, im”, lie without the circle OS towards 
SR is within the circle. ‘Therefore the distance OJ, in the case of the figure, is 
equal to — ax 2 (SP"—Sm" + )SM"—Sp"+SM"). - Lorre 
44. If the paths of rays P, M, p, m, be projected on we direction Os of the or- 
dinarily reflected ray, thelengths of their projections will be ate os? eyes = , 9 2 ,9 a ; 
respectively. The projections upon Os of the rays p, m, will be nee ays positive ; and 
the projections of the rays P, M, will be positive or negative according as the points 
P", M", lie above the point s or below it ; that is, according as the points P”, WM”, 
lie without the circle OS towards p and m, or within the circle. So that if SPmMps 
be a ray entering the crystal at O and emerging from the first surface at e, and if a 
perpendicular e7 be let fall from e upon Os, the distance Oz from the point O 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 = (—sP" + sm"—sM" + sp"), in 
the case of the figure. 
45. Let us imagine that the light in the incident ray S'O, instead of being inter- 
rupted at O by the crystal, had continued to move with the same velocity V in the 
same right line OS, leaving the point O at the moment when the refracted light 
enters the crystal at O. 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 reflections, we shall find that the emergent is behind the imaginary ray, and 
