in accordance with

_ _+[oo]
/ /
A'I0^2 = | | I^2 d[xi]d[eta],
_/_/-[oo]

the relation between A and A' is AA' = [lambda]^2f^2. Since A' is in
some sense the area of the diffraction pattern, it may be considered
to be a rough criterion of the definition, and we infer that the
definition of a point depends principally upon the area of the
aperture, and only in a very secondary degree upon the shape when the
area is maintained constant.

4. _Theory of Circular Aperture._--We will now consider the important
case where the form of the aperture is circular.

Writing for brevity

k[xi]/f = p, k[eta]/f = q, (1),

we have for the general expression (S 11) of the intensity

[lambda]^2f^2I^2 = S^2 + C^2 (2),

where
_ _
/ /
S = | | sin(px + qy)dx dy, (3),
_/_/
_ _
/ /
C = | | cos(px + qy)dx dy, (4).
_/_/

When, as in the application to rectangular or circular apertures, the
form is symmetrical with respect to the axes both of x and y, S = 0,
and C reduces to
_ _
/ /
C = | | cos px cos qy dx dy, (5).
_/_/

In the case of the circular aperture the distribution of light is of
course symmetrical with respect to the focal point p = 0, q = 0; and C
is a function of p and q only through [sqrt](p^2 + q^2). It is thus
sufficient to determine the intensity along the axis of p. Putting q =
0, we get
_ _ _+R
/ / / /
C = | | cos px dx dy = 2 | cos px \/(R^2 - x^2) dx,
_/_/ _/-R

R being the radius of the aperture. This integral is the Bessel's
function of order unity, defined by

_[pi]
z /
J1(z) = ---- | cos(z cos [phi]) sin^2 [phi] d[phi] (6).
[pi] _/0

Thus, if x = R cos [phi],

2J1(pR)
C = [pi]^2R ------- (7);
pR

and the illumination at distance r from the focal point is

/ 2[pi]Rr \
4J1^2( --------- )
[pi]^2R^4 \f[lambda]/
I^2 = ----------- . ----------------- (8).
[lambda]^2f^2 / 2[pi]Rr \^2
( --------- )
\f[lambda]/