ion
is not the imperfection of actual images so much as the possibility of
their being as good as we find them.

At the focal point ([xi] = 0, [eta] = 0) all the secondary waves agree
in phase, and the intensity is easily expressed, whatever be the form
of the aperture. From the general formula (2), if A be the _area_ of
aperture,

I0^2 = A^2/[lambda]^2f^2 (7).

The formation of a sharp image of the radiant point requires that the
illumination become insignificant when [xi], [eta] attain small
values, and this insignificance can only arise as a consequence of
discrepancies of phase among the secondary waves from various parts of
the aperture. So long as there is no sensible discrepancy of phase
there can be no sensible diminution of brightness as compared with
that to be found at the focal point itself. We may go further, and lay
it down that there can be no considerable loss of brightness until the
difference of phase of the waves proceeding from the nearest and
farthest parts of the aperture amounts to 1/4[lambda].

When the difference of phase amounts to [lambda], we may expect the
resultant illumination to be very much reduced. In the particular case
of a rectangular aperture the course of things can be readily
followed, especially if we conceive f to be infinite. In the direction
(suppose horizontal) for which [eta] = 0, [xi]/f = sin [theta], the
phases of the secondary waves range over a complete period when sin
[theta] = [lambda]/a, and, since all parts of the horizontal aperture
are equally effective, there is in this direction a complete
compensation and consequent absence of illumination. When sin [theta]
= 3/2[lambda]/a, the phases range one and a half periods, and there
is revival of illumination. We may compare the brightness with that in
the direction [theta] = 0. The phase of the resultant amplitude is the
same as that due to the central secondary wave, and the discrepancies
of phase among the components reduce the amplitude in the proportion

_+3/2[pi]
1 /
----- | cos [phi] d[phi]: 1,
3[pi] _/-3/2[pi]

or -2/3[pi]:1; so that the brightness in this direction is 4/9[pi]^2 of
the maximum at [theta] = 0. In like manner we may find the
illumination in any other direction, and it is obvious that it
vanishes when sin [theta] is any multiple of [lamba]/a.

The reason of the augmentation of