(11).
[pi] _/0 2^2 2^2.4^2 2^2.4^2.6^2

The value of C for an annular aperture of radius r and width dr is thus

dC = 2 [pi]J0 (p[rho]) [rho] d[rho], (12).

For the complete circle,

_ pR
2[pi] /
C = ----- | J0(z) zdz
p^2 _/0

2[pi] /p^2R^2 p^4 R^4 p^6 R^6 \
= ------ ( ------ - ------- + ----------- - ... )
p^2 \ 2 2^2.4^2 2^2.4^2.6^2 /

2J1(pR)
= [pi]R^2 . ------- as before.
pR

In these expressions we are to replace p by k[xi]/f, or rather, since
the diffraction pattern is symmetrical, by kr/f, where r is the
distance of any point in the focal plane from the centre of the
system.

The roots of J0(z) after the first may be found from

z .050561 .053041 .262051
---- = i - .25 + ------- - ---------- + ---------- ... (13),
[pi] 4i - 1 (4i - 1)^3 (4i - 1)^5

and those of J1(z) from

z .151982 .015399 .245835
---- = i + .25 - ------- + ---------- + ---------- ... (14),
[pi] 4i + 1 (4i + 1)^3 (4i + 1)^5

formulae derived by Stokes (_Camb. Trans._, 1850, vol. ix.) from the
descending series.[1] The following table gives the actual values:--

+---+--------------------+--------------------+
| | z | z |
| i | ---- for J0(z) = 0 | ---- for J1(z) = 0 |
| | [pi] | [pi] |
+---+--------------------+--------------------+
| 1 | 7655 | 1 2197 |
| 2 | 1 7571 | 2 2330 |
| 3 | 2 7546 | 3 2383 |
| 4 | 3 7534 | 4 2411 |
| 5 | 4 7527 | 5 2428 |
| 6 | 5 7522 | 6 2439 |
| 7 | 6 7519 | 7 2448 |
| 8 | 7 7516 | 8 2454 |
| 9 | 8 7514 | 9 2459 |
|10 | 9 7513 | 10 2463 |
+---+--------------------+--------------------+

In both cases the image of a mathematical point is thus a symmetrical
ring system. The greatest brightness is at the centre, where

dC = 2[pi][rho] d[rho], C = [pi]R^2.

For a certain distance ou