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.

346 UNDULATORY THEORY OF OPTICS.
after reflection are inclined* on opposite sides of the plane
of incidence. If i + i'=90, that is if the angle of incidence is the polarizing angle, the plane of polarization of
the reflected ray coincides with the plane of incidence: and
if i be further increased, 3 and a have the same signs.
These results have been verified by numerous observations
and careful measures of M. Arago, and Sir David Brewster.
PROP. 28. Light is incident on the internal surface of
glass at an angle equal to or greater than that of total
reflection; to find the intensity and nature of the reflected
ray.
133. The expressions in (128) and (129) become impossible. Yet there is a reflected ray, whatever be the
nature of the vibrations in the incident light. And on the
principle of vis viva the intensity of the reflected ray ought
to be equal to that of the incident ray, since there is no
refracted ray to consume a part of the vis viva. And
indeed in the last state of the expressions of (128) and (129)
before becoming impossible, that is when i'= 90o, each of
them becomes= 1. After this the expression for the coefficient of vibrations perpendicular to the plane of incidence
{putting gsini for sini', and /(-1).2/(,2sin'i-l) for cosi'}
becomes,u sin i cos i - sin i 5/(- 1) </(u2 sin2 i - ),u sin i cos i + sin i /(- 1) /(' sin'i - 1)'
or cos20 - /(-1). sin20,
where tan 0 = /(2 sini - 1)
Il cos i
and that for the coefficient of vibrations parallel to the
plane of incidence becomes
* The inclinations are considered to be on the same side when (supposing
for facility of conception the angle of incidence to be considerable) the upper
parts of both planes are on the same side of the plane of incidence.