rther apparatus (usually a
telescope) of high magnifying power.

In the above discussion it has been supposed that the ruling is
accurate, and we have seen that by increase of m a high resolving
power is attainable with a moderate number of lines. But this
procedure (apart from the question of illumination) is open to the
objection that it makes excessive demands upon accuracy. According to
the principle already laid down it can make but little difference in
the principal direction corresponding to the first spectrum, provided
each line lie within a quarter of an interval (a + d) from its
theoretical position. But, to obtain an equally good result in the
m^th spectrum, the error must be less than 1/m of the above amount.[7]

There are certain errors of a systematic character which demand
special consideration. The spacing is usually effected by means of a
screw, to each revolution of which corresponds a large number (e.g.
one hundred) of lines. In this way it may happen that although there
is almost perfect periodicity with each revolution of the screw after
(say) 100 lines, yet the 100 lines themselves are not equally spaced.
The "ghosts" thus arising were first described by G. H. Quincke
(_Pogg. Ann._, 1872, 146, p. 1), and have been elaborately
investigated by C. S. Peirce (_Ann. Journ. Math._, 1879, 2, p. 330),
both theoretically and experimentally. The general nature of the
effects to be expected in such a case may be made clear by means of an
illustration already employed for another purpose. Suppose two similar
and accurately ruled transparent gratings to be superposed in such a
manner that the lines are parallel. If the one set of lines exactly
bisect the intervals between the others, the grating interval is
practically halved, and the previously existing spectra of odd order
vanish. But a very slight relative displacement will cause the
apparition of the odd spectra. In this case there is approximate
periodicity in the half interval, but complete periodicity only after
the whole interval. The advantage of approximate bisection lies in the
superior brilliancy of the surviving spectra; but in any case the
compound grating may be considered to be perfect in the longer
interval, and the definition is as good as if the bisection were
accurate.

[Illustration:

| | | | | ( ( ( | | | | ) | (

FIG. 9.--