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.

362 UNDULATORY THEORY OF OPTICS.
2 r
a. sin - (vt - ),
perpendicular to the plane of first polarization. On entering
the crystal this is resolved into. 7r
a. cos qb. sin- (v t - o)
perpendicular to the principal plane (which produces the
Ordinary ray), and
7TT
a. sin (. sin - (vt -` )
parallel to the principal plane (which produces the Extraordinary ray). The former of these expressions may be
assumed to be true after the Ordinary ray has emerged from
the crystal, provided that we make the proper alteration in
the value of, or t: but then for the Extraordinary ray
we must, by (154), take the expression
a. sin. sin - (v t - x) + -.
If the rays entered the eye in this state, there would be no
variation of intensity in tle light coming in different directions through the crystal. For the intensity of the ordinary
wave = a2 cos2 p, and that of the extraordinary wave=a2 sin2 p,
the suin of which, or a2, represents the intensity of the
united waves (102) and this is constant. Now the analyzing
plate being applied, those resolved parts only of the vibrations are preserved which are perpendicular to the plane of
analyzation. That* furnished by the ordinary ray is
2,7
a.cos p. cos (p + a).sin (vt - ):
and that furnished by the extraordinary ray is
As the analyzing plate does not transmit to the eye the whole of the
vibrations perpendicular to its plane of polarization, we ought in strictness to
multiply these expressions, in this and similar investigations, by a constant.
The omission is of no consequence in comparing the intensities of different
parts of the image.