| C = | 2π | ∫ | pr | J0(z) zdz = | 2π | { | p²R² | − | p4R4 | + | p6R6 | − ...}= πR² · | 2J1(pR) | as before. |
| p² | 0 | p² | 2 | 2²·4² | 2²·4²·6² | pR |
In these expressions we are to replace p by kξ/ƒ, or rather, since the diffraction pattern is symmetrical, by kr/ƒ, 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 | = i − .25 + | .050561 | − | .053041 | + | .262051 | (13), |
| π | 4i − 1 | (4i − 1)³ | (4i − 1)5 |
and those of J1(z) from
| z | = i + .25 − | .151982 | + | .015399 | − | .245835 | (14), |
| π | 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:—
| i | z/π for J0(z) = 0 | z/π for J1(z) = 0 |
| 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 = 2πρ dρ, C = π R².