Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

364 UNDULATORY THEORY OF OPTICS.
measuring from the top to the right. By the reflection at
the analyzing plate, this course of the rays is inverted with
regard to up and down, while it is not altered with regard
to right and left: but then, as the eye is placed to receive
the light in the direction opposite to that in which we look
in studying figure 35, there is another inversion with regard
to right and left, but none with regard to up and down.
On the whole therefore, this ray comes from a point whose
apparent angular distance from a certain point through which
the rays pass parallel to the axis is i, which distance is
measured in a direction that makes the angle <p +a or q
with the plane of analyzation, measuring from the upper
part of the plane to the right. In the image presented to
the eye, i may be considered as a radius vector, and J the
angle that it makes with the upper part of the line
that represents the plane of analyzation. The brightness,
putting q\ for p + a, is
a {cos2 a - sin (2 - ).sins. sin -.
157. Let a = 900, or let the analyzing plane be in
the position in which no light is reflected without the interposition of the crystal. The expression becomes
a2. sin2 2 f. sin2 7.
This is 0, whatever be the value of X and of I, when
sin2 2 = 0: that is, when
b = O, or =900, or = 180~, or = 2700.
This shews that, whatever be the kind of light, there is a
black cross, passing through the point of the image formed
by the light that is parallel to the axis. For all intermediate values of p it vanishes only when
7r1
-= 0, TT, 27, &c., or I=0, X, 2X, &c.,
/ 2avX / 4avX / 6 avX
or sini=0, _/ -a- 4avX /6a -, &c.
( (c2-a )T (c2-a') " (c' -a) T'