on 
crystalline Reflexion and Refraction. 4 
medium to the corresponding sums in the other, we get the three conditions 
(7, cos @, + 7; cos 6) cos%, = 7, cos 0, cos 7, + 7; cos 6; cos 24, 
7, sin 6, + 7; sin 6; = 7, sin 6, + 7; sin 6), (34) 
(7, cos 6, — 7; cos 6) sin?, = 7, cos @, sin 7, + 74 cos 6; sin 7, 
A fourth condition is supplied by the first of the equations (31), in which equa- 
tion we have to write 
cos a, = sin 0, cosz, cos a; = — sin 6} cos?,, 
and to substitute similar expressions for cos a,, cos a). 
The right line OQ is perpendicular to the transversal 7, and to the ray OT. 
The cosines of the angles a,, f,, y, may therefore be found by means of the cosines 
of the angles which the transversal and the ray make with the axes of x,, 7, 2,. 
The cosines of the angles which the transversal +, makes with these axes are 
respectively 
cos 6, COS 2,, sin 6,, — cos 6, sin Z,. 
As the plane which passes through the ray and the wave-normal OS is per- 
pendicular to the transversal 7,, this plane makes with the plane of incidence an 
angle equal to 90° + @, or 90° — @,. Let asphere, having its centre at O, be 
intersected in the points S,, T, by the right lines OS, OT, and in the points 
X,, Y,, Z, by the axes of x, y,, z,3 and conceive the points T, and Y, to be at 
the same side of the plane of x, z,, the spherical angle T,S,X, being 90° + @., 
and the spherical angle T,S,Z, being 90° — 6,- Let « be the angle which the 
ray makes with the wave-normal. Then, the angles which the ray makes with 
the axes of coordinates being measured by the arcs T,X,, T,Y,, T,Z,, the cosines 
of these angles respectively are 
sin 7, cos € — sin @, cos 7, sin ¢, cos @, sin e, COS 7, COS € + sin 6, sin ¢, sin e. 
Hence, as the transversal is at right angles to the ray, we have, by Lemmal, 
cos a, = sinZ,sin e + sin 6, cos 7, cos €, cos B, = — cos 6, cos €, ae 
ano . . . v2 
COS Y, = COS 72, SIN € — SiN @, SIN 2, COS €, Gs) 
In like manner, putting e’ for the angle which the other refracted ray makes with 
its wave-normal, we have 
cos a, = sin7,sin e’ + sin 6,cos% cose, cos 6, = — cos 6 cos €, 
COS Y3; = cos 2; sin e& — sin 64 sin 7 cos &’. 
(36) 
