applied to the Wave Theory of Light. 251 
31. We shall consider particularly the case of a doubly refracting crystal, with two 
plane faces parallel to each other, and surrounded by a medium of the common kind” 
wherein the constant velocity is.V; supposing, for the sake of clearness, that the 
crystal refracts more powerfully than the surrounding medium, so that the velocities 
in the crystal are less than the velocity V. 
A ray S'O, falling on the first surface of the crystal at the point O, is partly re- 
flected according to the common law of reflection, and partly refracted. The two re- 
fracted rays pass on to the second surface, where each of them is divided by internal 
reflection into a pair, the two reflected pairs being parallel to each other; while the 
two emergent rays—one from each refracted ray—are parallel to each other and to 
the incident ray S’O. ‘The directions of the rays within the crystal are usually found 
by the following construction. j 
32. Describe a wave surface of the crystal, having its centre at O the point of in- 
cidence. By the nature of the wave surface, a right line O 7'U, drawn from the point 
O, will in general cut this surface in two points 7} U, on the same side of O; and a ray 
passing through the crystal in a direction parallel to OT'U willhave one of the two 
velocities represented by the radii O7;, OU, taking a line of a certain length / to repre- 
sent the uniform velocity V inthe external medium, With the centre O and a radius 
OS equal to this line & describe a sphere. As the velocities in the crystal are sup- 
posed to be less than V, the wave surface will lie wholly within this sphere. Let the 
plane of the figure (Fg. 7) be the plane of incidence, perpendicular to the parallel 
faces of the crystal, and intersecting the first face in the right line #A. Through the 
point S, where the incident ray S’O, produced through the crystal, cuts the surface of 
the sphere, draw SJ at right angles to OS and meeting F'A in the point ZA right 
line perpendicular to the plane of the figure, and passing through this point J, we shall 
call the right line Z. 
33. Through the right line J draw two planes touching the two sheets cf the wave 
surface, on the side remote from the incident light, in the points T, 7”, which will lie 
within the sphere (32); then the incident plane wave, perpendicular to OS, will be re- 
fracted into two plane waves parallel to these two tangent planes; and the linesO T, OT", 
will be the directions of the refracted rays along which the refracted waves are pro- 
pagated. The lengths 07, O7", represent the velocities with which the light moves 
along the rays ; and of course the normal velocities, which are the velocities of the 
S refracted waves, are represented by the perpendiculars OG, OJ, let fall from O on 
the two tangent planes at 7,7”. These two perpendiculars OG, OH, evidently lie 
in the plane of the figure; but the points T, 7”, in general, do not lie in this plane. 
34. Again, through the right line J draw two other planes touching the wave surface, at 
the side of the incident light, in the points ¢,¢’. The rays OT,0 7”, arriving at the second 
surface of the crystal, will each be divided by internal reflection into two rays parallel ( 
to Ot,Ot'; and these four reflected rays, arriving at the first surface, will each be divided, 
VOL. XVII. 3F 
