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J5(z)

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Originally appearing in Volume V08, Page 243 of the 1911 Encyclopedia Britannica.
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J5(z) = aio See also:cos (z cos 0) de=1—12+2? 24 4242 22.42.62+ . (11). The value of C for an See also:annular See also:aperture of See also:radius r and width dr' is For the See also: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 See also:pattern is symmetrical, by kr/f, where r is the distance of any point in the See also:focal See also:plane from the centre of the See also: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 See also:Stokes (Camb. Trans., 285o, vol. ix.) from the descending See also:series.' 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 See also:bright rings are necessarily of inferior brilliancy as compared with the central disk. The first dark See also:ring in the diffraction pattern of the complete circular aperture occurs when r/f=1.2197Xa/2R .... See also:Writing for brevity .

(6). Jl(z)/!2\ . ., (1+-18.16 See also:

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 See also:image' of a mathematical point is thus a symmetrical ring system. The greatest brightness is at the- centre, and the See also: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 See also:Sir W. See also:Hamilton in a memoir on " Fluctuating Functions," The ascending series for Jj(z). used by Sir G. B. See also:Airy (Camb. Trans., ( See also:Roy. Irish Trans., 584o.

(15). We may compare this with the corresponding result for a rectangular aperture of width a, See also:

Elf=x/a; and it appears that in consequence of the preponderance of the central parts, the See also:compensation in the See also:case of the circle does not set in at so small an obliquity as when the circle is replaced by a rectangular aperture, whose See also:side is equal to the See also: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 See also: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 See also:William See also:Herschel and others that the See also:definition of a See also:telescope is often improved by stopping off a See also:part of the central See also:area of the See also:object-See also:glass; but the See also:advantage to be obtained in this way is in no case See also:great, and anything like a reduction of the aperture to a narrow annulus is attended by a development of the See also: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 See also:Bessel's functions.

End of Article: J5(z)

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