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.

334 UNDULATORY THEORY OF OPTICS.
find the curve where the perpendicular on the tangent is
proportional to </(a2cos0 + c2sin20), 0 being the angle
made by the tangent with the axis of i. It is well known
that this is an ellipse, whose axes in the directions of z
and x are in the proportion of a: c. Consequently, to
discover the path of the extraordinary ray, we must suppose the waves produced by vibrations parallel to a principal
plane to diverge in the form of a spheroid of revolution
round a line parallel to the axis of z, and must suppose
the semi-axes of the spheroid parallel and perpendicular to
z to be represented by a and c: and must then proceed
as for common light. The radius of the sphere into which
the ordinary wave has diverged must at the same time be
represented by a.
114. It is easily seen that the motion of an extraordinary wave in the crystal is not generally perpendicular
to its front. For let AB, fig. 29, be an aperture through
which a small part of an extraordinary wave passes: CD
a line parallel to the axis of the crystal. Consider A, a,
b, c, &c. as the origins of equal spheroidal waves, the axes
of the waves being parallel to CD. It is plain that the
part between E and F is the only place in which the
waves strengthen each other, as at all points on both sides
of this they precede or follow each other by different
quantities, and therefore mutually destroy each other, while
between E and F the neighbouring waves meet in nearly
the same phase. The wave therefore will seem to travel
from AB to EF. The general rule therefore is this; describe a spheroid whose axis is parallel to the axis of the
crystal, and find the point of its surface where the tangent
plane is parallel to the front of the wave; then the motion
of the wave is parallel to the radius of that point.
115. The general construction for determining the path
of both rays is this. In fig. 30, let the plane of the paper
be the plane of incidence, BA' the projection of the surface
of the crystal, AB the front of a wave moving in the
direction A '. Let CD be the axis of the crystal, not
necessarily in the plane of the paper. While a part. of