J1 (Z)  _2 - 22.4+22.42.6 22.42.62.8 + . When z is great, we may employ the semi-convergent series , ~/ r2 1 3 1 _3.5.7.1.3 3.5.7.9.11.1.3.5 . I-'V .`zr) co*—170 8 z 8.16.24 (i) + 8.16.24.32.40 7 \i) s— • (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 co-ordinates, 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 .