in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus
| ω1 = | 1 | ∫∫∫ Z | d | ( | e−ikr | ) dx dy dz. |
| 4πb² | dy | r |
Since the dimensions of T are supposed to be very small in comparison with λ, 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
| ω1 = | TZ | · | y | · | d | ( | e−ikr | ) (13). |
| 4πb² | r | dr | r |
In like manner we find
| ω2 = − | TZ | · | x | · | d | ( | e−ikr | ) (14). |
| 4πb² | 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 ω, we have
| ω = | TZ | · | √(x² + y²) | · | d | ( | e−ikr | ) = | TZ sin φ | d | ( | e−ikr | ), |
| 4πb² | r | dr | r | 4πb² | dr | r |
φ denoting the angle between r and z. In differentiating e−ikr/r with respect to r, we may neglect the term divided by r² as altogether insensible, kr being an exceedingly great quantity at any moderate distance from the origin of disturbance. Thus
| ω = | −ik · TZ sin φ | · | e−ikr | (15), |
| 4πb² | r |