The first of equations (9) gives
ω3 = 0 (10).
For ω1, we have
| ω1 = | 1 | ∫∫∫ | dZ | e−ikr | dx dy dz (11), |
| 4πb² | dy | r |
where r is the distance between the element dx dy dz and the point where ω1 is estimated, and
k = n/b = 2π/λ (12),
λ 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
| ∫ | e−ikr | dZ | dy = [ Z | e−ikr | ] − ∫ Z | d | ( | e−ikr | ) dy, |
| r | dy | r | dy | r |