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.

268 UNDULATORY THEORY OF OPTICS.
making with the wall the same angle as CC", but on the
opposite side of the perpendicular: and the front of the
wave will be B'C'D', making with the wall the same angle
as BCD, but on the opposite side: the extent of vibration
&c. remaining as before. And this appears to justify us
sufficiently in the assertion that the waves of light may be
reflected in the sane manner. With regard to the smoothness of the reflecting surface, all that is necessary is that
the elevations or depressions do not exceed a fraction of X.
32. The following is a more independent method of
arriving at the same result, and is perhaps satisfactory. In
fig. 8 let ABC be the front of a wave going in the direction of AA'. As soon as each successive small portion of
this has reached the surface, we will consider it as causing
an agitation in the ether next in contact with the surface,
and will suppose that agitation to be the center of a spherical wave, diverging with the same velocity as the plane
wave {see the note to (24)}. Let us now consider the state
of things when A has reached A'. B has reached B' some
time before:' and would at this time have arrived at D if
not interrupted. Consequently it has diverged into a sphere
ab whose radius = B'D. C reached the surface at a still
longer time previous, and would at this time have reached
E: it has therefore diverged into a sphere cd whose radius
is CE. The same holds for every intermediate point. If
now we examine the nature of the front of the grand wave
formed by all these little waves, we see that it must be
Y = cosp. p (v t - cos a - y cos3 - z cos y)
+ cos p. <( {vt + ( - 2 c) cos a- y cosS - cos y},
Z = cos?..p (v t - x cos a -y cos p - z cos y)
+cos.~ {vt+ (x-2c) cosa-ycos3- cos y}.
The differential equations are satisfied, and the condition X=o when x=c is
also satisfied. The first terms of X, Y, Z, taken together, express the original
wave, whose direction makes angles a, P, y, with.v, y, z; the second express
the new or reflected wave, whose direction makes angles 1800-a, f, and y
with x, y, and z. This shews that the direction of reflection follows the law
commonly enounced as the law of reflection. The intensity of the reflected
wave is the same as that of the incident wave. This is the theory of oblique
echos.