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.

COURSE OF A WAVE AFTER REFLECTION. 275
parallel to the direction of the wave's motion, or perpendicular
to its front. In fig. 12 lit AD be one position of the wave,
A'D'a succeeding position, and so on. From Prop. 9 it appears that the front of each small part of the wave makes the
same angle with the surface after reflection as before, but
on the opposite side of the normal: and that consequently
the line representing the direction of the wave's motion, and
which is perpendicular to the front, makes the same angle
with the normal before and after reflection. As all the lines
representing the direction of motion of different points of the
wave are parallel to the axis of the paraboloid, those which
represent the direction of motion after reflection (by a well
known theorem) converge to F the focus. Consequently
the form of the wave, which by (39) is the surface to which
all these lines are nornals, is a spherical surface whose center
is F. Thus then at one time A'D' will be the front of the
wave: at a later time BC will be the form of that part which
is not reflected, and A"B, D"C, the form of those parts which
are reflected, the part incident at A' having been reflected
to A": at a still later time, bc will be the form of the part
not reflected, and A"'B'b, D"'C'c the form of the reflected
parts, the part incident at A' having been reflected to A"', and
that incident at B having been reflected to B', &c: and when
the whole has been reflected, all trace of the original form
of the wave will be lost, and the existing form will be only
a spherical surface of which F is the center. - The concave
spherical wave goes on towards F, contracting till it passes
through that point, when all the different small parts cross,
and then they form a diverging spherical wave of which F
is the center.
It is easily seen that an explanation of exactly the same
kind applies to the effects of refraction, the velocity of the
wave being supposed to be altered in a given ratio as in
Prop. 10, and the direction of the motion of each part of the
wave being always supposed perpendicular to that part of the
front.
41. We have explained the motion of the wave after
reflection or refraction as if the terminating edges of the
front of the wave did not cause any disturbance beyond the
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