zeta] + (a^2 - b^2) -------- = -Z |
dx /

It will now be convenient to introduce the quantities.[=omega]1,
[=omega]2, [=omega]3 which express the _rotations_ of the elements of
the medium round axes parallel to those of co-ordinates, in accordance
with the equations

d[xi] d[eta] d[eta] d[zeta]
[=omega]3 = ----- - ------, [=omega]1 = ------ - -------,
dy dx' dz dy

d[zeta] d[xi]
[=omega]2 = ------- - ----- (8).
dx dz

In terms of these we obtain from (7), by differentiation and
subtraction,

(b^2[Delta]^2 + n^2) [=omega]3 = 0 \
(b^2[Delta]^2 + n^2) [=omega]1 = dZ/dy > (9).
(b^2[Delta]^2 + n^2) [=omega]2 = -dZ/dx /

The first of equations (9) gives

[=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