z^-1 J1(z) = J0(z) - J1'(z);
so that
d d
z^-1J1^2(z) = 1/2 -- J0^2 - 1/2 -- J1^2(z),
dz dz
and
_z
/
2 | z^-1 J1^2(z)dz = 1 - J0^2(z) - J1^2(z) (21).
_/0
If r, or z, be infinite, J0(z), J1(z) vanish, and the whole
illumination is expressed by [pi]R^2, in accordance with the general
principle. In any case the proportion of the whole illumination to be
found outside the circle of radius r is given by
J0^2(z) + J1^2(z).
For the dark rings J1(z) = 0; so that the fraction of illumination
outside any dark ring is simply J0^2(z). Thus for the first, second,
third and fourth dark rings we get respectively .161, .090, .062,
.047, showing that more than 9/10ths of the whole light is
concentrated within the area of the second dark ring (_Phil. Mag._,
1881).