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 PROIDUCED 13Y BIAXAL CRYSTALS. 375
This shews that both hyperbolas are rectangular, and that
the asymptotes of one are parallel and perpendicular to the
plane of first polarization, and those of the other inclined to
them by a. The intensity of light in the brushes is
a COS2 a.
165. When 3 = 0, or = 900, tan 23 = 0, and the first
hyperbolas are changed into two straight lines, one in the
direction of FG, and the other perpendicular to it, passing
through C. Similarly when 3 = a, or = 90' + a, the second
hyperbolas are changed into a similar cross. Whatever be
the value of /, if a = O or = 90~, the two pairs of hyperbolas coincide: but the value a = gives for the intensity
a2, or the brush is bright: and the value a= 90~ gives for
the intensity 0, or the brush is black.
166. The nature of the rings is determined by the
variations of value of the last term
- sin 2. sin (2 + + 2a). sin2 -
When c > 0 < 90 - a, or > 90~ < 180- a,
or > 180~ < 270~-a, or > 270~ < 3600-a,
(the limits of which are determined by the hyperbolas already described) the brightness is greatest when
I=0, or =, or = 2X, &c.,
and least when
X 3X.
I=, -, &c.
2 2- -'
When / > 90~ - a < 90~, the brightness is greatest when
X $X
I =, -' &c. and least when = 0, X, 2X, &c.
The cases when a = 0 or a 90~ will be very easily investigated by the reader.