ion of
the secondary waves will be limited to distances of a few wave-lengths
only from the boundary of opaque and transparent parts.

3. _Fraunhofer's Diffraction Phenomena._--A very general problem in
diffraction is the investigation of the distribution of light over a
screen upon which impinge divergent or convergent spherical waves after
passage through various diffracting apertures. When the waves are
convergent and the recipient screen is placed so as to contain the
centre of convergency--the image of the original radiant point, the
calculation assumes a less complicated form. This class of phenomena was
investigated by J. von Fraunhofer (upon principles laid down by
Fresnel), and are sometimes called after his name. We may conveniently
commence with them on account of their simplicity and great importance
in respect to the theory of optical instruments.

If f be the radius of the spherical wave at the place of resolution,
where the vibration is represented by cos kat, then at any point M
(fig. 2) in the recipient screen the vibration due to an element dS of
the wave-front is (S 2)

dS
- ------------- sin k(at - [rho]),
[lambda][rho]

[rho] being the distance between M and the element dS.

Taking co-ordinates in the plane of the screen with the centre of the
wave as origin, let us represent M by [xi], [eta], and P (where dS is
situated) by x, y, z. Then

[rho]^2 = (x - [xi])^2 + (y - [eta])^2 + z^2, f^2 = x^2 + y^2 + z^2;

so that

[rho]^2 = f^2 - 2x[xi] - 2y[eta] + [xi]^2 + [eta]^2.

In the applications with which we are concerned, [xi], [eta] are very
small quantities; and we may take

/ x[xi] + y[eta]\
[rho] = f ( 1 - -------------- ).
\ f^2 /

At the same time dS may be identified with dxdy, and in the
denominator [rho] may be treated as constant and equal to f. Thus the
expression for the vibration at M becomes

_ _
1 / / / x[xi] + y[eta]\
- --------------- | | sin k ( at - f + -------------- ) dxdy (1);
[lambda]^2[f]^2 _/_/ \ f /

and for the intensity, represented by the square of the amplitude,

_ _ _ _
1 | / / x[xi] + y[eta] |^2
I^2 = ------------- | | | sin k -------------- dxdy |