J5 (z) = aio cos (z cos 0) de=1—12+2?
24
4242 22.42.62+ . (11).
The value of C for an annular aperture of radius r and width dr' is
For the complete circle,
C=2'r ItJo(z)zdz p2 4
p 1 p2R'—  _ I
2 2 22.4 +22.42.6 =~rR2.2JR) as before.
In these expressions we are to replace p ,by kW/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 Jo(z) after the first may be found from
i ' 25+ 4i
z '0506611 '05(4i30412 +(4i 2015)15
1)'26i=— "—and those of Jl(z) from
z '151982 '015399 '245835 a=i+'25— 4i+1 + 4i ip—(4i+I)b .
formulae derived by Stokes (Camb. Trans., 285o, vol. ix.) from the descending series.' The following table gives the actual values:
ilnfor •forJi(z)=0 i RfarJo(z)=0 =fforJi(z)=0
Jo(z)= 0
1 '7655 .1.2197 6 5'7522 62439
2 1'7571 2'2330 7 6'7519 7'2448
3 2.7546 3'2383 8 7.7516 82454
4 3'7534 4'2411 9 8'7514 92459
5 4.7527 52428 10 9'7513 102463
where
dC =2rp dp, C = irR2.
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=1.2197Xa/2R ....
Writing for brevity
. (6).
Jl(z)/!2\ . ., (1+18.16 sin .5 /1\2
3.5.7.9.1.3.5(1\ 4 8.16.24.32 lz +
thus
dC=2'7rJo(pp)pdp, (12).
(13)
. (14),
In both cases the image' of a mathematical point is thus a symmetrical ring system. The greatest brightness is at the centre,
and the illumination at distance r from the focal point is
4J 2~rRr1
a'R* _ ( X j
12 hzf2 Rr 2 . (8)
(2.r ' The descending series for Jo(z) appears to have been first given
f~` by Sir W. Hamilton in a memoir on " Fluctuating Functions,"
The ascending series for Jj(z). used by Sir G. B. Airy (Camb. Trans., ( Roy. Irish Trans., 584o.
(15).
We may compare this with the corresponding result for a rectangular aperture of width a,
Elf=x/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 letter case the first dark ring occurs at a much smaller obliquity, viz:
r/f ='7655 X a/2 R
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 objectglass; 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.'
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
Jo'(z) =—JI(z), • • (16) ;
End of Article: J5 (z) 

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