282 
of Classes G and K are included. We may now 
answer decisively, and in the affirmative, the first two 
questions which were put. a few moments ago. 
Some stars actually have densities quite as low as 
any that might be required to explain the great 
brightness of the reddest giant stars; and these stars 
of low density show a very marked preference for 
the ‘‘later’’ spectral classes, while practically all 
the stars of ‘‘earlier’’ type are far denser. 
We can answer the third question as well, and in 
a quantitative fashion, if we are willing to assume 
that the eclipsing binaries, and also the telescopic 
double stars, of the various spectral classes are 
typical of the stars of these classes as a_ whole. 
Though this may not be rigorously true, there is 
good reason to believe that it is not seriously in error. 
We find, from Fig. 4, that the fifty eclipsing stars of 
Class A, if they all had the sun’s mass and surface 
brightness, but their own densities, would, on the 
average, be of the absolute magnitude 3:06. Now, 
referring to Fig. 3, we find that the mean absolute 
magnitude which the 115 visual double stars of Class 
AO there recorded would have, if they were equal in 
mass only to the sun, but had their own surface 
brightness as well as density, would be 1-07. The 
only difference between these two groups (if they are 
both typical of the stars of Class A in general) is 
that one has been reduced by computation to the 
sun’s surface brightness, while the other has not. 
It is therefore evident that the stars of Class A must, 
on the average, for equal surfaces, be two magnitudes 
brighter than the sun. Apart from the uncertainty 
whether the two groups compared are exactly typical, 
the probable error of this determination should be 
less than one-tenth of a magnitude. 
In similar fashion, we find that the mean absolute 
magnitude of fifty-two visual pairs of spectra Oeces 
to Bg, reduced to the sun’s mass, is —o-4, while that 
of twelve eclipsing binaries of similar spectrum, 
reduced to the sun’s mass, and surface brightness, is 
+2-8, which makes the surface brightness of an 
average star of Class B greater by 3-2m. than that 
of the sun. Again, for the stars of Class F, we get 
+26 for the mean reduced absolute magnitude of the 
sixty-nine visual pairs, and +3-7 for that of the nine 
eclipsing pairs, the difference of 1-1m. being approxi- 
mately the effect of surface brightness (somewhat 
more uncertain here, on account of the apparently 
different proportion of giant stars in the two groups). 
It appears, therefore, that in passing down the 
spectral series from B to G, the surface brightness 
of the stars decreases by about one magnitude from 
each class to the next; and we have previously found 
that. among the dwarf stars, the decrease in surface 
brightness in passing from G to M must be at least 
2} magnitudes more. Ail this has been shown with- 
out making any use whatever of the physical mean- 
ing of the spectra, which have simply been used as 
svmbols in classifying the stars into groups. The 
results are obviously in accordance with the view 
that the differences of spectral type arise from differ- 
ences of temperatur2. Indeed, they constitute new 
and important evidence in its favour. How well they 
agree with other independent lines of evidence is 
shown by comparing the relative surface brightness 
just computed with the colour-index for the various 
classes. Taking A as a standard, we have :— 
Spectrum B A F G K M 
Surface bright- 
ness aile2) (OO) OLO 2250) —— ans (at least) 
Colour-index... -0°3 0°O +0°3 +0°7 4+1'°2 +1°6 
The computed differences in surface brightness are 
NO. 23245 MOL. 03] 
NATURE 
[May 14, 1914 
in all cases about three times the colour-indices, in 
good agreement with the theoretical ratio. 
We may now estimate the density of the redder 
giant stars. It appears from Fig. 3 that the mean 
absolute magnitude of the giant stars, if reduced to 
the sun’s mass, is +0-6 for Class G. +0-5 for Class k, 
and oo for Classes K5 and M. The differences 
between these values are small, and we may well take 
the general mean, +0-44, as typical of the whole. 
This corresponds to about fifty times the sun's 
luminosity. Such a giant star of Class G, if of the 
sun’s surface brightness, would have to be of about 
seven times the sun’s radius, and of 1/350 of its 
density. If we assume, on the basis of the foregoing 
study of the dwarf stars, that the surface intensities 
of the giant stars of Classes K and M are respectively 
I'5 and 3 magnitudes fainter than that of the sun, 
we find that their densities must be 1/2800 and 
1/25,000 of the sun’s density. The observed densities 
of several eclipsing variables of Classes G and K are 
of just the order of magnitude here found, so that 
there is direct observational evidence in favour of all 
our conclusions, except the very low density assigned 
to the giant stars of Class M (among which no 
eclipsing variables have yet been found, so that their 
densities cannot be directly determined). But there 
is nothing improbable about so low a density, for we 
know of at least one star—W Crucis—the density of 
which is still smaller. 
Before leaving these diagrams we should notice 
that, by comparing the data of Fig. 3 with those of 
Figs. 1 and 2, we may obtain the average masses of 
the stars of the various types. Consider, for example, 
the stars of spectra B to B5. From Fig.-3 we see 
that, if these stars were reduced to the sun’s mass 
without changing either their surface brightness or 
density, their mean absolute magnitude would be 
—o-6. But the actual mean absolute magnitude of 
the stars of this spectral class is —2-0 according to 
Campbell, or —o-8 according to Boss. Taking the 
mean of these determinations, we find that these stars 
are, on the average, 2-1 times as bright as stars of 
unit mass, but of the same surface brightness and 
density, would be, from which it follows that their 
average surface area must be 2-1 times that of the 
latter stars, and their average mass 3-0 times that of 
the sun. The uncertainty whether the groups of 
stars which we are comparing are really exactly 
similar is here more serious than usual; if Campbell’s 
stars are taken as typical, the mean mass comes out 
seven times that of the sun. It should be noticed 
that the ‘‘average’’ mass here obtained corresponds 
approximately to the average of the logarithms of the 
individual masses, and hence to their geometrical 
mean, which will be somewhat smaller than their 
arithmetical mean, and that we are here dealing with 
the mass of the brighter component of each system 
only. For the twelve spectroscopic binaries of spec- 
trum B, which are available for comparison, the mean 
mass of the brighter components is about 9, and the 
geometrical mean probably about 7-5, times the sun’s 
mass. As the observational selection in this case 
undoubtedly favours the larger masses, there is no 
serious discrepancy between the two results. 
Proceeding similarly for the stars of the other 
spectral classes, we obtain the results collected in 
Table VII. The observed absolute magnitudes of the 
stars in clusters have been taken in preference to 
those of the stars of directly measured parallax, for 
the reasons already stated, and for the giant stars the 
mean of the results of Boss and Campbell has been 
used (except for Class G, for which Boss’s value alone 
really represents them). 
