J5(z)

This article appears in Volume V08, Page 243 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: INV-JED

J5(z) = aio cos (z cos 0) de=112+2?
24
4242 22.42.62+ . (11).
The value of C for an annular aperture of

radius

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

SYSTEM

.
The roots of Jo(z) after the first may be found from
i ' 25+ 4i-
z '0506611 '05(4i-30412 +(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

SERIES (a Latin word from serere, to join)

.' The following table gives the actual values:
i-lnfor 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

BRIGHT, JOHN (1811-188g)
BRIGHT, SIR CHARLES TILSTON

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

SYSTEM

. The greatest brightness is at the- centre,
and the

illumination

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

SERIES (a Latin word from serere, to join)

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

DIAMETER (from the Cr. &a, through, µErpov, measure)

of the circle.
Again, if we compare the complete circle with a narrow annular aperture of the same

radius

RADIUS

, we see that in the

letter

LETTER (through Fr. lettre from Lat. littera or litera, letter of the
alphabet
- ALPHABET (see also WRITING)
; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

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

PART

of the central area of the object-glass; but the advantage to be obtained in this way is in no case

great

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

DIAMETER (from the Cr. &a, through, µErpov, measure)

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) ;

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