[oo]
converges.

It may be instructive to contrast this with the case of an infinitely
narrow annular aperture, where the brightness is proportional to
J0^2(z). When z is great,

/ 2
J0(z) = \ / ----- cos(z^-1/4 [pi]).
\/ [pi]z

The mean brightness varies as z^-1; and the integral
_
/ [oo]
| J0^2(z)z dz is not convergent.
_/ 0

5. _Resolving Power of Telescopes._--The efficiency of a telescope is of
course intimately connected with the size of the disk by which it
represents a mathematical point. In estimating theoretically the
resolving power on a double star we have to consider the illumination of
the field due to the superposition of the two independent images. If the
angular interval between the components of a double star were equal to
twice that expressed in equation (15) above, the central disks of the
diffraction patterns would be just in contact. Under these conditions
there is no doubt that the star would appear to be fairly resolved,
since the brightness of its external ring system is too small to produce
any material confusion, unless indeed the components are of very unequal
magnitude. The diminution of the star disks with increasing aperture was
observed by Sir William Herschel, and in 1823 Fraunhofer formulated the
law of inverse proportionality. In investigations extending over a long
series of years, the advantage of a large aperture in separating the
components of close double stars was fully examined by W. R. Dawes.

The resolving power of telescopes was investigated also by J. B. L.
Foucault, who employed a scale of equal bright and dark alternate parts;
it was found to be proportional to the aperture and independent of the
focal length. In telescopes of the best construction and of moderate
aperture the performance is not sensibly prejudiced by outstanding
aberration, and the limit imposed by the finiteness of the waves of
light is practically reached. M. E. Verdet has compared Foucault's
results with theory, and has drawn the conclusion that the radius of the
visible part of the image of a luminous point was equal to half the
radius of the first dark ring.

The application, unaccountably long delayed, of this principle to the
microscope by H. L. F. Helmholtz in 1871 is the foundation of the
important doctrine of the _microscopic limit_. It is true that in 1823
Fraunhofer, inspired by his observations upon gratings, had