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.

378 UNDULATORY THEORY 0F OPTICS.
There is a brush interrupting the rings where
sin(2cj + 2 a) = 0:
this is the same equation as that which determines the second
hyperbolas in (164), and which when 3 = a or = 90~ + a
becomes a cross. When sin (29 + 2a) is positive, the inX 7X
tensity is maximum if I = - -, &c., and minimum if
4
X 5X
I=-, -i, &c.: and the contrary when sin(20 + 2a) is
negative. These spaces are separated by the brush: consequently the bright rings on one side of the brush correspond
to the dark rings on the other side. The form of the rings
is just the same as in (167).
PROP. 37. A plate of uniaxal or biaxal crystal, cut in
any direction different from those of Prop. 31 and 34, is
placed between the polarizing and analyzing plates: to find
the appearance presented to the eye.
172. The general expression for the brightness in (155),
- o + s.cos2.cos(2( +2a) + sin2(. sin(20 + 2a). cos --
applies to this case. To confine ourselves to the most important instances we will make a = 90~, which reduces the
expression to
a sin20 (1 - cos --
By I is meant here the space that one ray (which whether
in uniaxal or biaxal crystals we shall call the Ordinary ray)
is retarded more than the Extraordinary ray, and to which
the two expressions in (162) still apply, observing only that
in the former i' is the angle made with the axis. The essential difference between this case and that of Propositions
31 and 34 is, that here I is large for all rays which pass
nearly perpendicular to the plate.
(I) If the plate be thick, all traces of colours will disappear (as we have seen in several cases of interference where one ray had gained many multiples