[lambda]^2f^2 |_ _/_/ f _|
_ _ _ _
1 | / / x[xi] + y[eta] |^2
+ ------------- | | | cos k -------------- dxdy | (2).
[lambda]^2f^2 |_ _/_/ f _|

This expression for the intensity becomes rigorously applicable when f
is indefinitely great, so that ordinary optical aberration disappears.
The incident waves are thus plane, and are limited to a plane aperture
coincident with a wave-front. The integrals are then properly
functions of the _direction_ in which the light is to be estimated.

In experiment under ordinary circumstances it makes no difference
whether the collecting lens is in front of or behind the diffracting
aperture. It is usually most convenient to employ a telescope focused
upon the radiant point, and to place the diffracting apertures
immediately in front of the object-glass. What is seen through the
eye-piece in any case is the same as would be depicted upon a screen
in the focal plane.

Before proceeding to special cases it may be well to call attention to
some general properties of the solution expressed by (2) (see Bridge,
_Phil. Mag._, 1858).

If when the aperture is given, the wave-length (proportional to k^-1)
varies, the composition of the integrals is unaltered, provided [xi]
and [eta] are taken universely proportional to [lambda]. A diminution
of [lambda] thus leads to a simple proportional shrinkage of the
diffraction pattern, attended by an augmentation of brilliancy in
proportion to [lambda]^-2.

If the wave-length remains unchanged, similar effects are produced by
an increase in the scale of the aperture. The linear dimension of the
diffraction pattern is inversely as that of the aperture, and the
brightness at corresponding points is as the _square_ of the area of
aperture.

If the aperture and wave-length increase in the same proportion, the
size and shape of the diffraction pattern undergo no change.

We will now apply the integrals (2) to the case of a rectangular
aperture of width a parallel to x and of width b parallel to y. The
limits of integration for x may thus be taken to be -1/2a and +1/2a,
and for y to be -1/2b, +1/2b. We readily find (with substitution for k
of 2[pi]/[lambda])

[pi]a[xi] [pi]b[eta]