from crystallized Surfaces. 107 
ZR=p,PR=gq. Then we shall find from equation (4.), 
after various substitutions and reductions, 
272 
& = K cos* g (cos* $—cos*p); where K = (2 3 eres (5) 
In deducing this value of 8, the approximations were made 
with a tacit reference to the case of reflexion in air from a 
common rhomb of Iceland spar. The coefficient K, in this 
case, is equal to about nine degrees, and the resulting nume- 
rical values of the polarizing angles in various azimuths agree 
very well with your experiments. You will perceive that the 
value of 3 is the same in supplementary azimuths, which ex- 
plains the observation, cited in the beginning of my letter, re- 
lative to the equality of the polarizing angles at opposite sides 
of the perpendicular I Z in a given plane of incidence. 
When the point R falls upon O, we have 8 = 0, and7+4 
equal to a right angle. Hence, when the cotangent of ZR 
is equal to the ordinary index, the tangent of the polarizing 
angle is equal to the same index. This theorem, though de- 
duced from an approximate equation, might be shown to be 
exact. 
When the axis of the crystal lies in the plane of incidence, 
we may obtain an exact expression for the polarizing angle. 
The condition of polarization then becomes 
a af 2__},2) cin dl 1 APR Bs) jl 
cos (2+ ¢')—(a°—0*) sin p' cos h ayia Age 
from which, by the proper substitutions, we obtain the fol- 
lowing expression : 
0; (6.) 
- 9. 1—a® cos? A—B? sin? A 
sin? z.= a BP : (7.) 
where A denotes the complement of Z P, or the inclination of 
the axis to the face of the crystal, and z is the polarizing angle. 
This formula, in a shape somewhat different, was communi- 
cated, above a year ago, to Professor Lloyd, who has noticed 
it, in connexion with your paper, in his Report on Physical 
Optics. When a and become equal, the formula gives your 
law of the tangent for ordinary media. 
The foregoing results show that, when a ray is polarized 
by reflexion from a crystal, the plane of polarization deviates 
from the plane of incidence, except when the axis lies in the 
latter plane; and that the deviation may be made very great 
by placing the crystal in contact with a medium whose 
refractive power is nearly equal to that of the crystal it- 
self; for when z is nearly equal to ¢ or to 4, the divisor sin 
(t—¢) or sin ({(—9’) is TG small, and therefore tan 6 or 
2 
