emarkable bands are seen under certain
conditions when a tolerably pure spectrum is regarded with the naked
eye, or with a telescope, _half the aperture being covered by a thin
plate_, e.g. _of glass or mica_. The view of the matter taken by the
discoverer (_Phil. Mag._, 1837, 10, p. 364) was that any ray which
suffered in traversing the plate a retardation of an odd number of half
wave-lengths would be extinguished, and that thus the spectrum would be
seen interrupted by a number of dark bars. But this explanation cannot
be accepted as it stands, being open to the same objection as Arago's
theory of stellar scintillation.[9] It is as far as possible from being
true that a body emitting homogeneous light would disappear on merely
covering half the aperture of vision with a half-wave plate. Such a
conclusion would be in the face of the principle of energy, which
teaches plainly that the retardation in question leaves the aggregate
brightness unaltered. The actual formation of the bands comes about in
a very curious way, as is shown by a circumstance first observed by
Brewster. When the retarding plate is held on the side towards the red
of the spectrum, _the bands are not seen_. Even in the contrary case,
the thickness of the plate must not exceed a certain limit, dependent
upon the purity of the spectrum. A satisfactory explanation of these
bands was first given by Airy (_Phil. Trans._, 1840, 225; 1841, 1), but
we shall here follow the investigation of Sir G. G. Stokes (_Phil.
Trans._, 1848, 227), limiting ourselves, however, to the case where the
retarded and unretarded beams are contiguous and of equal width.

The aperture of the unretarded beam may thus be taken to be limited by
x = -h, x = 0, y = -l, y= +l; and that of the beam retarded by R to be
given by x = 0, x = h, y= -l, y = +l. For the former (1) S 3 gives
_ _
1 / 0 / +l / x[xi] + y[eta]\
- --------- | | sin k (at - f + -------------- )dxdy
[lambda]f _/-h _/-l \ f /

2lh f k[eta]l 2f k[xi]h / [xi]h \
= - --------- . ------- sin ------- . ------ sin ------ . sin k (at - f - ----- ) (1),
[lambda]f k[eta]l f k[xi]h 2f \ 2f /

on integration and reduction.

For the retarded stream the only difference is that we must subtract R
from at, and that the