482 
ASTRONOMY: F. H. SHARES 
Cm^ depends upon the absolute magnitude M; provisionally, at 
least, it must also be regarded as a function of the spectrum. Differ- 
ences in absolute brightness for the same spectral type doubtless are 
accompanied by differences in stellar dimensions, and hence also by 
variations in the extent of the stellar atmosphere. These may produce 
modifications of intensity in the continuous spectrum, which possibly 
are also dependent upon the spectral type. 
The last term C^r, provides for a scattering of Hght in its passage 
through space. It is a function of the parallax tt or the distance A, 
but is independent of the spectrum. It may be expressed in the 
form adopted by Kapteyn, namely, 
c, = d^ (3) 
in which A == O'M/tt, while d is the change in the color index produced 
by an increase of one unit in the distance. 
Finally, for the more precise definition of the color symbols it is con- 
venient to suppose that the observed color index C is connected, with the 
color classes h, a, J, etc., by the relation 
C = OAOs ' (4) 
or 
s = 2.5C, (5) 
in which 5 is specifically denoted by bO, aO, JO, etc., when its numerical 
values are —1, 0, +1, etc., respectively. The relation is analogous to 
that connecting the average color index with spectral type. 
Equation (1) can therefore be written in the form 
s = 2.5 C = S 2.5 Cm,s + 2.5 dA, (6) 
from which it appears that s differs from the spectrum S only in so far 
as the object in question deviates in luminosity and distance from the 
mean conditions underlying the color index and spectrum relation de- 
fined by equation (2). The color symbols b, aj, etc. are thus intimately 
related to the spectrum symbols B, A, F, etc., and it will often be con- 
venient to refer to them as hypothetical spectra. 
As a matter of further notation, we may define the difference C — Cs 
= Ce as the color excess, that is, the excess of the observed color over 
the mean color index corresponding to the spectrum S. 
We have 
Ce = OA(s - S) = CM,s-\-dA. (7) 
The color excess therefore represents the combined effect of the luminos- 
ity and distance terms, and is useful in investigations undertaken for 
the determination of the reahty and magnitude of the phenomena giving 
rise to these terms. 
