which becomes on putting for u, v, and [alpha] their values, and
putting
/ f k[eta]l \^2
( ------- sin ------- ) = Q (3),
\k[eta]l f /
_ _
4l^2 [pi][xi]h | / 2[pi]R 2[pi][xi]h\ |
I = Q . ------------ sin^2 --------- |2 + 2 cos ( -------- - ---------- ) | (4).
[pi]^2[xi]^2 [lambda]f |_ \[lambda] [lambda]f / _|
If the subject of examination be a luminous line parallel to [eta], we
shall obtain what we require by integrating (4) with respect to [eta]
from -[oo] to +[oo]. The constant multiplier is of no especial
interest so that we may take as applicable to the image of a line
_ _
2 [pi][xi]h | / 2[pi]R 2[pi][xi]h \ |
I = ------ sin^2 --------- |1 + cos ( -------- - ---------- ) | (5).
[xi]^2 [lambda]f |_ \[lambda] [lambda]f / _|
If R = 1/2[lambda], I vanishes at [xi]= 0; but the whole illumination,
represented by
_
/ +[oo]
| I d[xi], is independent of the value of R. If R = 0,
_/-[oo]
1 2[pi][xi]h
I = ------ sin^2 ----------,
[xi]^2 [lambda]f
in agreement with S 3, where a has the meaning here attached to 2h.
The expression (5) gives the illumination at [xi] due to that part of
the complete image whose geometrical focus is at [xi] = 0, the
retardation for this component being R. Since we have now to integrate
for the whole illumination at a particular point O due to all the
components which have their foci in its neighbourhood, we may
conveniently regard O as origin. [xi] is then the co-ordinate
relatively to O of any focal point O' for which the retardation is R;
and the required result is obtained by simply integrating (5) with
respect to [xi] from -[oo] to +[oo]. To each value of [xi] corresponds
a different value of [lambda], and (in consequence of the dispersing
power of the plate) of R. The variation of [lambda] may, however, be
neglected in the integration, except in 2[pi]R/[lambda], where a small
variation of [lambda] entails a comparatively large alteration of
phase. If we write
[rho] = 2[pi]R/[lambda] (6),
we must regard [rho] as a function of [xi], and we may take with
sufficient approximation under any ordinary circumstances
[rho] = [rho]' + [=omega][xi] (7),
where [rho]' denotes the value of [rho] at O, and [=omega] is a
constant, which is positive when the retarding plate is held at the
side on which the lue of the spectrum _is seen_. The possibility of
dark bands depends upon [=omega] being positive. Only in this case can
cos {[rho]' + ([=omega] - 2[pi]h/[lambda]f)[xi]}
retain the constant value -1 throughout the integration, and then only
when
[=omega] = 2[pi]h / [lambda]f (8)
and
cos [rho]' = -1 (9).
The first of these equations is the condition for the formation of
dark bands, and the second marks their situation, which is the same
as that determined by the imperfect theory.
The integration can be effected without much difficulty. For the first
term in (5) the evaluation is effected at once by a known formula. In
the second term if we observe that
cos {[rho]' +([=omega] - 2[pi]h/[lambda]f)[xi]} = cos {[rho]'- g1[xi]}
= cos [rho]' cos g1[xi] + sin [rho]' sin g1[xi],
we see that the second part vanishes when integrated, and that the
remaining integral is of the form
_+[oo]
/ d[xi]
w = | sin^2 h1[xi] cos g1[xi] ------,
_/-[oo] [xi]^2
where
h1 = [pi]h/[lambda]f, g1 = [omega] - 2[pi]h/[lambda]f (10).