the Double Refraction of Quartz. 463 
When ¢=0, the light passes along the axis of the crystal. In this case we have 
"je =1, and k= +1; which shows that there are two rays, circularly polarized in oppo- 
site directions. The value of s for each ray may be had from equation (5.) or (6.), 
by putting +1 and —1 successively for &. Calling these values s’ and s”, we find 
‘ C ' aC 
s*=a'—2r7, YSe ere (9.) 
C aC 
2 9 SP Bay en . 
s=a+2n—, s"=a (1 ay) (10.) 
Suppose a plate of quartz to have two parallel faces perpendicular to the axis, and 
conceive a ray of light, polarized in a given plane, to fall perpendicularly on it. The 
incident rectilinear vibration may be resolved into two opposite circular vibrations, 
which will pass through the crystal with different velocities ; and which, after their 
emergence into air, will again compound a rectilinear vibration, whose direction wil] 
make a certain angle p with that of the incident vibration : so that the plane of polari- 
zation will appear to have been turned round through an angle equal to p, called the 
- angle of rotation. This angle may be determined by means of the preceding formule. 
Putting @ for the thickness of the crystalline plate, the circularly polarized wave whose 
velocity is s’, will pass through it in the time 
68 ( aC ) : 
al 
and the wave whose velocity is s", in the time 
— 
s a 
ees salt) aC 
7 ( ead ). 
Therefore, if ° be the difference of the times, we have 
QrCd 
P s= =T" (11.) 
But, since the velocity of propagation in air is supposed to be unity, the time and 
the space described are represented by the same quantity ; and therefore 0, which is 
evidently a line, denotes the distance between the fronts of the two circularly polarized 
waves, when they emerge into air. ‘The waves being at this distance from each other, 
if we conceive, at the same depth in each of them, a molecule performing its circular 
vibration, and carrying a radius of its circle along with it, the two radii will revolve 
in contrary directions, and will always cross each other in a position parallel to the 
incident rectilinear vibration. Now let two series of such waves be superposed, so as 
to agitate every molecule by their compound effect, and it is evident, that, when 
the radius vector of one component vibration attains the position just mentioned, the 
radius vector of the other will be separated from it by an angle equal to =, where 
VOL. XVII. es 4M : 
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