J1 (Z) _2  22.4+22.42.6 22.42.62.8 + . When z is great, we may employ the semiconvergent series
,
~/ r2 1 3 1 _3.5.7.1.3 3.5.7.9.11.1.3.5.
I'V .`zr) co*ā170 8 z 8.16.24 (i)
+ 8.16.24.32.40 7 \i) sā ā¢ (10).
A table of the values of 2z'Jl(z) has been given by E. C. J. Lommel (Schlomilch, 187o, 15, p. 166), to whom is due the first systematic application of Bessel's functions to the diffraction integrals.
The illumination vanishes in correspondence with the roots of the equation Jl(z) =o. If these be called z2, z3, ... the radii of the dark rings in the diffraction pattern are
'zi f1z2
2xR ' ark' ' '
being thus inversely proportional to R.
The integrations may also be effected by means of polar coordinates, taking first the integration with respect to r"so as to obtain the result for an infinitely thin annular aperture. Thus, if
x=p cos y=p sin (p,
C = ff cos px dx dy = f R f02 r cos (pp cos 0) pdp do. Now by definition
End of Article: J1 (Z) 

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