volume T, we may
write

TZ y d / e^-ikr \
[=omega]1 = -------- . - . -- ( ------ ) (13).
4[pi]b^2 r dr \ r /

In like manner we find

TZ x d / e^-ikr \
[=omega]2 = -------- . - . -- ( ------- ) (14).
4[pi]b^2 r dr \ r /

From (10), (13), (14) we see that, as might have been expected, the
rotation at any point is about an axis perpendicular both to the
direction of the force and to the line joining the point to the source
of disturbance. If the resultant rotation be [omega], we have

TZ [sqrt](x^2 + y^2) d /e^-ikr\
[=omega] = ------- . ----------------- . -- ( ------ ) =
4[pi]b^2 r dr \ r /

TZ sin[phi] d /e^-ikr\
= ----------- -- ( ------ ),
4[pi]b^2 dr \ r /

[phi] denoting the angle between r and z. In differentiating
e^(-ikr)/r with respect to r, we may neglect the term divided by r^2 as
altogether insensible, kr being an exceedingly great quantity at any
moderate distance from the origin of disturbance. Thus

-ik.TZ sin[phi] /e^-ikr\
[=omega] = --------------- . ( ------ ) (15),
4[pi]b^2 \ r /

which completely determines the rotation at any point. For a
disturbing force of given integral magnitude it is seen to be
everywhere about an axis perpendicular to r and the direction of the
force, and in magnitude dependent only upon the angle ([phi]) between
these two directions and upon the distance (r).

The intensity of light is, however, more usually expressed in terms of
the actual displacement in the plane of the wave. This displacement,
which we may denote by [zeta]', is in the plane containing z and r,
and perpendicular to the latter. Its connexion with [=omega]is
expressed by [=omega] = d[zeta]'/dr; so that

TZ sin [phi] /e^-ikr\
[zeta]' = ----------- . ( ------ ) (16),
4[pi]b^2 \ r /

where the factor e^int is restored.

Retaining only the real part of (16), we find, as the result of a
local application of force equal to

DTZ cos nt (17),

the disturbance expressed by

TZ sin [phi] /cos(nt - kr)\
[zeta]' = --------