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 outwards this remains sensibly unimpaired and
then gradually diminishes to zero, as the secondary waves become
discrepant in phase. The subsequent revivals of brightness forming the
bright rings are necessarily of inferior brilliancy as compared with
the central disk.
The first dark ring in the diffraction pattern of the complete
circular aperture occurs when
r/f = 1.2197 X [lambda]/2R (15).
We may compare this with the corresponding result for a rectangular
aperture of width a,
[xi]/f =[lambda]/a;
and it appears that in consequence of the preponderance of the central
parts, the compensation in the case of the circle does not set in at
so small an obliquity as when the circle is replaced by a rectangular
aperture, whose side is equal to the diameter of the circle.
Again, if we compare the complete circle with a narrow annular
aperture of the same radius, we see that in the latter case the first
dark ring occurs at a much smaller obliquity, viz.
r/f = .7655 X [lambda]/2R.
It has been found by Sir William Herschel and others that the
definition of a telescope is often improved by stopping off a part of
the central area of the object-glass; but the advantage to be obtained
in this way is in no case great, and anything like a reduction of the
aperture to a narrow annulus is attended by a development of the
external luminous rings sufficient to outweigh any improvement due to
the diminished diameter of the central area.[2]
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);
1
J2(z) = - J1(z) - J1'(z) (17);
z
2
J0(z) + J2(z) = - J1(z) (18).
z