| u = U − y = U − U−1 − | 2 | U−3 − | 13 | U−5 − | 146 | U−7 − ... (6). |
| 3 | 15 | 105 |
In the first quadrant there is no root after zero, since tan u > u, and in the second quadrant there is none because the signs of u and tan u are opposite. The first root after zero is thus in the third quadrant, corresponding to m = 1. Even in this case the series converges sufficiently to give the value of the root with considerable accuracy, while for higher values of m it is all that could be desired. The actual values of u/π (calculated in another manner by F. M. Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.
Since the maxima occur when u = (m + ½)π nearly, the successive values are not very different from
| 4 | , | 4 | , | 4 | , &c. |
| 9π² | 25π | 49π² |
The application of these results to (3) shows that the field is brightest at the centre ξ = 0, η = 0, viz. at the geometrical image of the radiant point. It is traversed by dark lines whose equations are
ξ = mfλ / a, η = mfλ / b.
Within the rectangle formed by pairs of consecutive dark lines, and not far from its centre, the brightness rises to a maximum; but these subsequent maxima are in all cases much inferior to the brightness at the centre of the entire pattern (ξ = 0, η = 0).
By the principle of energy the illumination over the entire focal plane must be equal to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from ξ = ∞ to ξ = +∞, and from η = −∞ to η = +∞ should be equal to ab. This integration, employed originally by P. Kelland (Edin. Trans. 15, p. 315) to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula
| ∫ | +∞ | sin²u | du = ∫ | +∞ | sin u | du = π. |
| −∞ | u² | −∞ | u |
It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a²b². If the aperture be increased, not only is the total brightness over the focal plane increased with it, but there is also a concentration of the diffraction pattern. The form of (3) shows immediately that, if a and b be altered, the co-ordinates of any characteristic point in the pattern vary as a−1 and b−1.