he ratio a
: a + d; and, except for the difference in illumination, the
appearance of a line of light is the same as if the aperture were
perfectly free.

The lateral (spectral) images occur in such directions that the
projection of the element (a + d) of the grating upon them is an exact
multiple of [lambda]. The effect of each of the n elements of the
grating is then the same; and, unless this vanishes on account of a
particular adjustment of the ratio a : d, the resultant amplitude
becomes comparatively very great. These directions, in which the
retardation between A and B is exactly mn[lambda], may be called the
principal directions. On either side of any one of them the
illumination is distributed according to the same law as for the
central image (m = 0), vanishing, for example, when the retardation
amounts to (mn [+-] 1)[lambda]. In considering the relative brightnesses
of the different spectra, it is therefore sufficient to attend merely
to the principal directions, provided that the whole deviation be not
so great that its cosine differs considerably from unity.

We have now to consider the amplitude due to a single element, which
we may conveniently regard as composed of a transparent part a bounded
by two opaque parts of width 1/2d. The phase of the resultant effect is
by symmetry that of the component which comes from the middle of a.
The fact that the other components have phases differing from this by
amounts ranging between [+-] am[pi]/(a + d) causes the resultant
amplitude to be less than for the central image (where there is
complete phase agreement). If Bm denote the brightness of the m^th
lateral image, and B0 that of the central image, we have

_ _+ am[pi]/(a + d) _
| / 2am[pi] |^2 /a + d \^2 am[pi]
B_m : B0 = | | cosx dx :- ------- | = ( ------ ) sin^2 ------ (1).
|_ _/ a + d _| \am[pi]/ a + d
-am[pi]/(a + d)

If B denotes the brightness of the central image when the whole of the
space occupied by the grating is transparent, we have

B0 : B = a^2 : (a + d)^2,

and thus

1 am[pi]
Bm : B = --------- sin^2 ------ (2).
m^2[pi]^2 a + d

The sine of an angle can never be greater than unity; and consequently
under the most fav