ormula (5) still holds, and, if the
deviation be reckoned from the direction of the regularly reflected
rays, it is expressed as before by ([theta] + [phi]), and is a minimum
when [theta] = [phi], that is, when the diffracted rays return upon
the course of the incident rays.

[Illustration: FIG. 7.]

In either case (as also with a prism) the position of minimum
deviation leaves the width of the beam unaltered, i.e. neither
magnifies nor diminishes the angular width of the object under view.

From (5) we see that, when the light falls perpendicularly upon a
grating ([theta] = 0), there is no spectrum formed (the image
corresponding to m = 0 not being counted as a spectrum), if the
grating interval [sigma] or (a + d) is less than [lambda]. Under these
circumstances, if the material of the grating be completely
transparent, the whole of the light must appear in the direct image,
and the ruling is not perceptible. From the absence of spectra
Fraunhofer argued that there must be a microscopic limit represented
by [lambda]; and the inference is plausible, to say the least (_Phil.
Mag._, 1886). Fraunhofer should, however, have fixed the microscopic
limit at 1/2[lambda], as appears from (5), when we suppose [theta] =
1/2[pi], [phi] = 1/2[pi].

[Illustration: FIG. 8.]

We will now consider the important subject of the resolving power of
gratings, as dependent upon the number of lines (n) and the order of
the spectrum observed (m). Let BP (fig. 8) be the direction of the
principal maximum (middle of central band) for the wave-length
[lambda] in the m^th spectrum. Then the relative retardation of the
extreme rays (corresponding to the edges A, B of the grating) is
mn[lambda]. If BQ be the direction for the first minimum (the darkness
between the central and first lateral band), the relative retardation
of the extreme rays is (mn + 1)[lambda]. Suppose now that [lambda] +
[delta][lambda] is the wave-length for which BQ gives the principal
maximum, then

(mn + 1)[lambda] = mn([lambda] + [delta][lambda]);

whence

[delta][lambda]/[lambda] = 1/mn (6).

According to our former standard, this gives the smallest difference
of wave-lengths in a double line which can be just resolved; and we
conclude that the resolving power of a grating depends only upon the
total number of lines, and upon the order of the spectrum, without
regard to