and
| 2 ∫ | z | z-1J1²(z)dz = 1 − J0²(z) − J1²(z) (21). |
| 0 |
If r, or z, be infinite, J0(z), J1(z) vanish, and the whole illumination is expressed by πR², in accordance with the general principle. In any case the proportion of the whole illumination to be found outside the circle of radius r is given by
J0²(z) + J1²(z).
For the dark rings J1(z) = 0; so that the fraction of illumination outside any dark ring is simply J0²(z). Thus for the first, second, third and fourth dark rings we get respectively .161, .090, .062, .047, showing that more than 9⁄10ths of the whole light is concentrated within the area of the second dark ring (Phil. Mag., 1881).
When z is great, the descending series (10) gives
| 2J1(z) | = | 2 | √( | 2 | ) sin(z − ¼π) (22); |
| z | z | πz |
so that the places of maxima and minima occur at equal intervals.
The mean brightness varies as z-3 (or as r-3), and the integral found by multiplying it by zdz and integrating between 0 and ∞ converges.
It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to J0²(z). When z is great,