The maximum brightnesses and the places at which they occur are easily determined with the aid of certain properties of the Bessel’s functions. It is known (see [Spherical Harmonics]) that
J0′(z) = −J1(z), (16);
J2(z) = (1/z) J1(z) − J1′(z) (17);
J0(z) + J2(z) = (2/z) J1(z) (18).
The maxima of C occur when
| d | ( | J1(z) | ) = | J1′(z) | − | J1(z) | = 0; |
| dz | z | z | z² |
or by 17 when J2(z) = 0. When z has one of the values thus determined,
| 2 | J1(z) = J0(z). |
| z |
The accompanying table is given by Lommel, in which the first column gives the roots of J2(z) = 0, and the second and third columns the corresponding values of the functions specified. If appears that the maximum brightness in the first ring is only about 1⁄57 of the brightness at the centre.
| z | 2z−1J1(z) | 4z−2J1²(z) |
| .000000 | +1.000000 | 1.000000 |
| 5.135630 | − .132279 | .017498 |
| 8.417236 | + .064482 | .004158 |
| 11.619857 | − .040008 | .001601 |
| 14.795938 | + .027919 | .000779 |
| 17.959820 | − .020905 | .000437 |