ial present by which light could be absorbed,
and the effect depends upon a difference of retardation in passing the
alternate parts. It is possible to prepare gratings which give a
lateral spectrum brighter than the central image, and the explanation
is easy. For if the alternate parts were equal and alike transparent,
but so constituted as to give a relative retardation of 1/2[lambda], it
is evident that the central image would be entirely extinguished,
while the first spectrum would be four times as bright as if the
alternate parts were opaque. If it were possible to introduce at every
part of the aperture of the grating an arbitrary retardation, all the
light might be concentrated in any desired spectrum. By supposing the
retardation to vary uniformly and continuously we fall upon the case
of an ordinary prism: but there is then no diffraction spectrum in the
usual sense. To obtain such it would be necessary that the retardation
should gradually alter by a wave-length in passing over any element of
the grating, and then fall back to its previous value, thus springing
suddenly over a wave-length (_Phil. Mag._, 1874, 47, p. 193). It is
not likely that such a result will ever be fully attained in practice;
but the case is worth stating, in order to show that there is no
theoretical limit to the concentration of light of assigned
wave-length in one spectrum, and as illustrating the frequently
observed unsymmetrical character of the spectra on the two sides of
the central image.[4]

We have hitherto supposed that the light is incident perpendicularly
upon the grating; but the theory is easily extended. If the incident
rays make an angle [theta] with the normal (fig. 6), and the
diffracted rays make an angle [phi] (upon the same side), the relative
retardation from each element of width (a + d) to the next is (a + d)
(sin[theta] + sin[phi]); and this is the quantity which is to be
equated to m[lambda]. Thus

sin[theta] + sin[phi] = 2 sin 1/2([theta] + [phi]) cos 1/2([theta] - [phi]) = m[lambda]/(a + d) (5).

The "deviation" is ([theta] + [phi]), and is therefore a minimum when
[theta] = [phi], i.e. when the grating is so situated that the angles
of incidence and diffraction are equal.

In the case of a reflection grating the same method applies. If
[theta] and [phi] denote the angles with the normal made by the
incident and diffracted rays, the f