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.

RINGS PRODUCED BY UNIAXAL CRYSTALS. 367
For, the succession of rings depends on the alteration of
Tr1
values of sin2-r: and this is removed by the evanescence
of its multiplier. This equation gives
- 0, or = 900 or = 1800 or =2700, or = a, or = 900 + a
or = 1800 +a, or = 270~ + a.
Consequently there are two rectangular crosses, inclined a to
each other, which interrupt the rings: and the intensity of
the light in these crosses is a2 cos2a. For the parts between
= 0, J = a, or between g\ = 900, \ = 90~ + a, or between,f=180~, Jf=180~+a, or between N-=270~, =,=270~+a, the
multiplier of sin2 is positive, and the light is therefore
greatest when & and least en
greatest when I=-7 = —, &c., and least when I=X,
=2X, &c.: these four sectors are therefore occupied by
portions of rings nearly similar to those in (157), the intensity for the portions of the bright rings being
a2
-{1 + cos (4x - 2a) or a2 cos2 (2 - a),
and that for the portions of the darker rings a cos2 a.
But for the parts between f = a, \i = 90~, &c., the multiplier
of sin2 - is negative: the light is least when I = &c.,
and greatest when I= X, 2X, &c.: these sectors therefore
are occupied by portions of rings nearly similar to those
in (158), the intensity of the portions of the bright rings
being a2 cos' a, and that of the fainter rings
a2 cos2 (2! - a).
The brighter rings in the last mentioned sectors have the
same radii and the same brightness as the darker rings in
those first mentioned: and this brightness is the same as
the brightness in the eight rays of the crosses.