tion of the width of the central band in the image of
a luminous line depends upon discrepancies of phase among the
secondary waves, and since the discrepancy is greatest for the waves
which come from the edges of the aperture, the question arises how far
the operation of the central parts of the aperture is advantageous. If
we imagine the aperture reduced to two equal narrow slits bordering
its edges, compensation will evidently be complete when the projection
on an oblique direction is equal to 1/2[lambda], instead of [lambda] as
for the complete aperture. By this procedure the width of the central
band in the diffraction pattern is halved, and so far an advantage is
attained. But, as will be evident, the bright bands bordering the
central band are now not inferior to it in brightness; in fact, a band
similar to the central band is reproduced an indefinite number of
times, so long as there is no sensible discrepancy of phase in the
secondary waves proceeding from the various parts of the _same_ slit.
Under these circumstances the narrowing of the band is paid for at a
ruinous price, and the arrangement must be condemned altogether.

A more moderate suppression of the central parts is, however,
sometimes advantageous. Theory and experiment alike prove that a
double line, of which the components are equally strong, is better
resolved when, for example, one-sixth of the horizontal aperture is
blocked off by a central screen; or the rays quite at the centre may
be allowed to pass, while others a little farther removed are blocked
off. Stops, each occupying one-eighth of the width, and with centres
situated at the points of trisection, answer well the required
purpose.

It has already been suggested that the principle of energy requires
that the general expression for I^2 in (2) when integrated over the
whole of the plane [xi], [eta] should be equal to A, where A is the
area of the aperture. A general analytical verification has been given
by Sir G. G. Stokes (_Edin. Trans._, 1853, 20, p. 317). Analytically
expressed--

_ _+[oo] _ _
/ / / /
| | I^2 d[xi]d[eta] = | | dxdy = A (9).
_/_/-[oo] _/_/

We have seen that I0^2 (the intensity at the focal point) was equal to
A^2/[lambda]^2f^2. If A' be the area over which the intensity must be I0^2
in order to give the actual total intensity