n which it is propagated with
respect to the front of the primary wave.

If, as suffices for all practical purposes, we limit the application
of the formulae to points in advance of the plane at which the wave is
supposed to be broken up, we may use simpler methods of resolution
than that above considered. It appears indeed that the purely
mathematical question has no definite answer. In illustration of this
the analogous problem for sound may be referred to. Imagine a flexible
lamina to be introduced so as to coincide with the plane at which
resolution is to be effected. The introduction of the lamina (supposed
to be devoid of inertia) will make no difference to the propagation of
plane parallel sonorous waves through the position which it occupies.
At every point the motion of the lamina will be the same as would have
occurred in its absence, the pressure of the waves impinging from
behind being just what is required to generate the waves in front. Now
it is evident that the aerial motion in front of the lamina is
determined by what happens at the lamina without regard to the cause
of the motion there existing. Whether the necessary forces are due to
aerial pressures acting on the rear, or to forces directly impressed
from without, is a matter of indifference. The conception of the
lamina leads immediately to two schemes, according to which a primary
wave may be supposed to be broken up. In the first of these the
element dS, the effect of which is to be estimated, is supposed to
execute its actual motion, while every other element of the plane
lamina is maintained at rest. The resulting aerial motion in front is
readily calculated (see Rayleigh, _Theory of Sound_, S 278); it is
symmetrical with respect to the origin, i.e. independent of [theta].
When the secondary disturbance thus obtained is integrated with
respect to dS over the entire plane of the lamina, the result is
necessarily the same as would have been obtained had the primary wave
been supposed to pass on without resolution, for this is precisely the
motion generated when every element of the lamina vibrates with a
common motion, equal to that attributed to dS. The only assumption
here involved is the evidently legitimate one that, when two systems
of variously distributed motion at the lamina are superposed, the
corresponding motions in front are superposed also.

The method of resolut