---- . ( ------------ ) (18).
4[pi]b^2 \ r /

The occurrence of sin [phi] shows that there is no disturbance
radiated in the direction of the force, a feature which might have
been anticipated from considerations of symmetry.

We will now apply (18) to the investigation of a law of secondary
disturbance, when a primary wave

[zeta] = sin(nt - kx) (19)

is supposed to be broken up in passing the plane x = 0. The first step
is to calculate the force which represents the reaction between the
parts of the medium separated by x = 0. The force operative upon the
positive half is parallel to OZ, and of amount per unit of area equal
to

-b^2D d[zeta]/dx = b^2kD cos nt;

and to this force acting over the whole of the plane the actual motion
on the positive side may be conceived to be due. The secondary
disturbance corresponding to the element dS of the plane may be
supposed to be that caused by a force of the above magnitude acting
over dS and vanishing elsewhere; and it only remains to examine what
the result of such a force would be.

Now it is evident that the force in question, supposed to act upon the
positive half only of the medium, produces just double of the effect
that would be caused by the same force if the medium were undivided,
and on the latter supposition (being also localized at a point) it
comes under the head already considered. According to (18), the effect
of the force acting at dS parallel to OZ, and of amount equal to

2b^2kD dS cos nt,

will be a disturbance

dS sin [phi]
[zeta]' = ------------ cos(nt - kr) (20),
[lambda]r

regard being had to (12). This therefore expresses the secondary
disturbance at a distance r and in a direction making an angle [phi]
with OZ (the direction of primary vibration) due to the element dS of
the wave-front.

The proportionality of the secondary disturbance to sin [phi] is
common to the present law and to that given by Stokes, but here there
is no dependence upon the angle [theta] between the primary and
secondary rays. The occurrence of the factor [lambda]r^-1, and the
necessity of supposing the phase of the secondary wave accelerated by
a quarter of an undulation, were first established by Archibald Smith,
as the result of a comparison between the primary wave, supposed to