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.

CIRCULARLY AND ELLIPTICALLY POLARIZED LIGHT. 351
(2) Let a have any value, S being= 90~. (This is
the general case of Fresnel's rhomb). Here
27rI. 27 -y = a sin a. cos (v t - ), z =a cos a. sin- (vt - v)
X x
y2 z2
and + ~ = 1.
ar sins a a" cos" a
Tliat is, every particle describes an ellipse, whose
semi-axes are a sina parallel to the plane of reflection, and a cos a perpendicular to that plane.
(3) In the general case, a and ~ having any values,
y = a sin a sin (vt -,). cos $ + cos - (vt - ). sin },
x X
O7r
and z = a cos a. cos -- (v t - ).
À
277z
Hence cos -(vt - ) =X a cos a
and
(y - tan a. sin. z)2 = a2 sin2 a.cos2. 1 - cos2 (vt - )
= a2 sin2.cos2. - tan2 a.cos2. z2:
the equation to an ellipse whose axes are inclined
to the plane of reflection.
(4) If we compare the expressions for y and z in the
first case with the equations to a circular helix, (t
being considered constant) we find that they exactly
coincide. That is, a series of particles which were
originally in a straight line, will be at any subsequent time in the form of a circular helix. In the
other cases, the position of the particles will be
what may by analogy be called an elliptic helix.
(5) For all values of i, if a= 0, or if a =900, the
reflected light has the same polarization as the
incident light.