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.

REFLECTION WHEN POLARIZED LIGHT FALLS ON GLASS. 345
in the direction making with the plane of incidence an angle
(90~- a):
and this may be resolved into
2wr
a cos a. (v t - )
perpendicular to the plane of incidence, and
27r
a. sin a. sin - (vt x)
À
parallel to the plane of incidence. And these expressions
will apply to the reflected ray, giving x the same alteration in both, and altering the coefficients in the ratios de-.termined in (128) and (129). Hence we shall have after
reflection,
Displacement perpendicular to the plane of incidence
sin (i'-i), 2r
a cos a..., sin (v t - ).
sin (i + i) \Displacement parallel to the plane of incidence
tan (i' - i). 2 -
- a sin a. sin (vt - x).
tan (i'+ i) ÀX
Since these are in the saine ratio whatever be the value of
w, it follows that the displacement compounded of these is
entirely in one plane, and therefore the reflected light is
polarized. And if,3 is the angle at which the new plane
of polarization is inclined to the plane of incidence, or
90~- [3 the angle at which the new direction of vibration
is inclined to the plane of incidence, we have
sin (i'- i)
a cos a. sin (i' + i) cos (i'- )
cot = -- - - cot a..~,.tan (i'- i) cos (i' + i)
- a sin a.
tan (i' + i)
cos (i' + i)
or tan j =-tan a.
cos (i'- i)
When i and i' are both small,,3 and a have different
signs: this shews that the planes of polarization before and