by the mean
wave-length. This method is especially used with a so-called
_objective-grating_, which consists of a series of metallic threads,
stretched parallel to each other at equal intervals. On account of the
diffraction of the light we now get in the focal plane of the objective,
with the use of these gratings, not only a fainter image of the star at
the place where it would have arisen without grating, but also at both
sides of this image secondary images, the distances of which from the
central star are certain theoretically known multiples of the effective
wave-lengths. In this simple manner it is possible to determine the
effective wave-length, and this being a tolerably well-known function of
the spectral-index, the latter can also be found. This method was first
proposed by HERTZSPRUNG and has been extensively used by BERGSTRAND,
LUNDMARK and LINDBLAD at the observatory of Upsala and by others.

15. _Colour-index._ We have already pointed out in Sec.9 that the colour
may be identified with the mean wave-length ([lambda]_0). As further
[lambda]_0 is closely connected with the spectral index (_s_), we may
use the spectral index to represent the colour. Instead of _s_ there may
also be used another expression for the colour, called the colour-index.
This expression was first introduced by SCHWARZSCHILD, and is defined in
the following way.

We have seen that the zero-point of the photographic scale is chosen in
such a manner that the visual magnitude _m_ and the photographic
magnitude _m'_ coincide for stars of spectral index 0.0 (A0). The
photographic magnitudes are then unequivocally determined. It is found
that their values systematically differ from the visual magnitudes, so
that for type B (and O) the photographic magnitudes are smaller than the
visual, and the contrary for the other types. The difference is greatest
for the M-type (still greater for the N-stars, though here for the
present only a few determinations are known), for which stars if amounts
to nearly two magnitudes. So much fainter is a red star on a
photographic plate than when observed with the eye.

_The difference between the photographic and the visual magnitudes is
called the colour-index (_c_)._ The correlation between this index and
the spectral-index is found to be rather high (_r_ = +0.96). In L. M.
II, 19 I have deduced the following tables giving the spectral-type
corresponding to a given colour-index, and inversely.

TABLE