[=omega]3 = 0 (10).
For =[omega]1, we have
_ _ _ -ikr
1 / / / dZ e
[=omega]1 = -------- | | | -- ----- dx dy dz (11),
4[pi]b^2 _/_/_/ dy r
where r is the distance between the element dx dy dz and the point
where [=omega]1 is estimated, and
k = n/b = 2[pi]/[lambda] (12),
[lambda] being the wave-length.
(This solution may be verified in the same manner as Poisson's
theorem, in which k = 0.)
We will now introduce the supposition that the force Z acts only
within a small space of volume T, situated at (x, y, z), and for
simplicity suppose that it is at the origin of co-ordinates that the
rotations are to be estimated. Integrating by parts in (11), we get
_ -ikr _ _ _
/ e dZ | Ze^-ikr | / d / e^-ikr\
| ------ -- dy = | ------- | - | Z -- ( ------- ) dy,
_/ r dy |_ r _| _/ dy \ r /
in which the integrated terms at the limits vanish, Z being finite
only within the region T. Thus
_ _ _ -ikr
1 / / / d /e^ \
[=omega]1 = ------- | | | Z -- ( -------- ) dx dy dz.
4[pi]b^2 _/_/_/ dy \ r /
Since the dimensions of T are supposed to be very small in comparison
with [lambda], the factor d/dy (e^-ikr / r) is sensibly constant; so
that, if Z stand for the mean value of Z over the volume T, we may
write
TZ y d / e^-ikr \
[=omega]1 = -------- . - . -- ( ------ ) (13).
4[pi]b^2 r dr \ r /