t 1/57 of the
brightness at the centre.

+-------------------------------------------+
| z 2z^-1 J1(z) 4z^-2 J1^2(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 |
+-------------------------------------------+

We will now investigate the total illumination distributed over the
area of the circle of radius r. We have

[pi]^2R^4 4J1^2(z)
I^2 = ------------- . ------- (19),
[lambda]^2f^2 z^2

where

z = 2[pi]Rr/[lambda]f (20).

Thus
_ _ _
/ [lambda]^2f^2 / /
2[pi] | I^2rdr = ------------- | I^2zdz = [pi]R^2.2 | z^-1 J1^2(z)dz.
_/ 2[pi]R^2 _/ _/

Now by (17), (18)

z^-1 J1(z) = J0(z) - J1'(z);

so that

d d
z^-1J1^2(z) = 1/2 -- J0^2 - 1/2 -- J1^2(z),
dz dz

and

_z
/
2 | z^-1 J1^2(z)dz = 1 - J0^2(z) - J1^2(z) (21).
_/0

If r, or z, be infinite, J0(z), J1(z) vanish, and the whole
illumination is expressed by [pi]R^2, 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

J0^2(z) + J1^2(z).

For the dark rings J1(z) = 0; so that the fraction of illumination
outside any dark ring is simply J0^2(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 - 1/4[pi]) (22);
z z \/ \[pi]z/

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