any other considerations. It is here of course assumed that
the n lines are really utilized.

In the case of the D lines the value of [delta][lambda]/[lambda] is
about 1/1000; so that to resolve this double line in the first
spectrum requires 1000 lines, in the second spectrum 500, and so on.

It is especially to be noticed that the resolving power does not
depend directly upon the closeness of the ruling. Let us take the case
of a grating 1 in. broad, and containing 1000 lines, and consider the
effect of interpolating an additional 1000 lines, so as to bisect the
former intervals. There will be destruction by interference of the
first, third and odd spectra generally; while the advantage gained in
the spectra of even order is not in dispersion, nor in resolving
power, but simply in brilliancy, which is increased four times. If we
now suppose half the grating cut away, so as to leave 1000 lines in
half an inch, the dispersion will not be altered, while the brightness
and resolving power are halved.

There is clearly no theoretical limit to the resolving power of
gratings, even in spectra of given order. But it is possible that, as
suggested by Rowland,[5] the structure of natural spectra may be too
coarse to give opportunity for resolving powers much higher than those
now in use. However this may be, it would always be possible, with the
aid of a grating of given resolving power, to construct artificially
from white light mixtures of slightly different wave-length whose
resolution or otherwise would discriminate between powers inferior and
superior to the given one.[6]

If we define as the "dispersion" in a particular part of the spectrum
the ratio of the angular interval d[theta] to the corresponding
increment of wave-length d[lambda], we may express it by a very simple
formula. For the alteration of wave-length entails, at the two limits
of a diffracted wave-front, a relative retardation equal to
mnd[lambda]. Hence, if a be the width of the diffracted beam, and
d[theta] the angle through which the wave-front is turned,

ad[theta] = mn d[lambda],

or dispersion = mn/a (7).

The resolving power and the width of the emergent beam fix the optical
character of the instrument. The latter element must eventually be
decreased until less than the diameter of the pupil of the eye. Hence
a wide beam demands treatment with fu