crystalline Reflexion and Refraction. 4] 
while the positive direction of z{ lies within the angle made by the positive direc- 
tion of x, and the negative direction of z,. Let 7, be the angle of incidence, and 
i,, 7, the angles of refraction ; then 
Z, = ZS 2, + 2, COS 2, Z, = Z, Sint, — Z, COS2, 
Zy = @, Sint, + 2, COS 2, 2 = 2, sin 1, + Z, cos %. 
(28) 
These values are to be written in the expressions (25). They show that the 
phases, and therefore the displacements, are independent of y,. 
Since the conditions relative to the plane of x, y, must hold at every instant 
of time, and for every point of that plane, the coefficients of ¢, as well as those of 
xy, in the values of the different phases, must be identical; so that we must have 
ua Se 45, sin?,__sin?, __sin2, 
Mi he Ae Sneha, © (29) 
Therefore, when z, = 0, the supposition 
Lv yo (30) 
renders the phases identical, independently of ¢ and ,. And, from the form of 
the equations of condition, it is easy to see that this supposition is necessary ; 
because the equations (20), when the values (26) are substituted in them, contain 
only the cosines of the phases ; and the equations (21), when the values (27) are 
substituted in them, contain only the sines of the phases. Making the latter sub- 
stitution, and attending to the relations just mentioned, we find 
7, COS a, + 7; Cos a = rz, cos a, + 7’7; COs a}, 
7, cos B, + 7; cos B, = 77, cos B, + 7’73 COs 3. 
(31) 
In these equations, the angles by whose cosines each transversal is multi- 
plied are the angles which a plane, passing through the directions of that trans- 
versal and of the corresponding ray, makes with the planes of y, z, and x, 2. 
This is evident with regard to the incident and reflected rays. And if we refer 
to the diagram in the preceding section, it will also be evident with regard to 
the refracted rays ; for OQ is perpendicular to the transversal r,, and to the right 
line OT, which is the direction of the corresponding ray. 
Taking O for the point of incidence, let right lines proceeding from it 
VOL. XXI. G 
