May 7, 1914] 
NATURE 
259 
we find, for the mean absolute magnitudes of all the 
stars of each class, the following values :— 
TABLE V. 
Mean Absolute Magnitudes. 
Stars of measured parallax Stars in clusters 
Spectrum no “Abs. mag. Formula O-C No. Abs.mag. Formula O-C 
m. m. 
Ba — —_ — —= 20 See eT — Or 
B8 — _— = — $8 +03 +02 +0'! 
Ao 6 +14 +1.4 oOo 13 o'5 06 -O'! 
A4 7 225 2°3 | +052 «26 Tie7. Tas) Gt Ona 
‘Fo — — = aS 2°4 227. = O33 
iP ee) 3,7 O'S — Pe = 
Be: 15a =a = ae 7 SiS SiG o"0 
Pa Come 9 4°5 .—-0°2 =) ee a = 
' F8 8 Bal bea 0%) 6 a5 4°2 44 -0'2 
Go 29 5°7 56 +01 18 50 48 +02 
ae es «8906-00. Oa So 07 
Biko - 28 7px 777-06 9g 6°4 6°9 -0O'5 
K4 19 92 8°06 -FO'6) 7 eyaOme-t a7 — 087) 
Ma 10 +99 +68 +01. — —_ _— — 
- The rate of decrease of brightness with increasing 
redness is very nearly the same for the stars with 
directly measured parallaxes and the stars in clusters, 
but the latter appear, with remarkable - consistency, 
to be about o-8m. brighter than the former. This 
seems at first sight very puzziing,. but it is un- 
doubtedly due to the way in which the stars observed 
for parallax were selected. Most observers, in pre- 
paring their working lists, have included mainly those 
stars’ which were brighter than a given magnitude 
and had proper-motions exceeding some definite limit. 
Of the stars above this limiting magnitude, those of 
greater actual luminosity will be, on the average, 
farther away, and have smaller proper-motions, than 
those of small luminosity, and selection by proper- 
motion favours the latter. The limitation of our 
present lists to stars the parallaxes of which have 
been determined with a probable error not exceeding 
42 per cent. of their own amounts, though necessary 
to diminish the effects of casual errors of observation, 
works in the same direction, for, among the stars of 
any given visual magnitude, those of greatest 
luminosity have the smallest parallaxes, and are least 
likely to pass the test. The difference shown in our 
table need not therefore alarm us, but it is clear that 
the stars in clusters, rather than the others, should 
be taken as typical of the dwarf stars as a whole. 
For both sets of stars the absolute magnitude appears 
to be very nearly a linear function of the spectral 
class (if B is regarded as 1, A as 2, etc.) The 
columns headed ‘formula’? in Table V. give the 
values calculated from the expressions M=1-4m.+ 
21m. (Sp.—A) for the stars of directly measured 
parallax, and M=o-6m.+2-:1m. (Sp.—A) for the stars 
in clusters. The residuals from these empirical 
formule, for the mean absolute magnitudes of the 
observed stars of different classes, average +0-33m. 
in the first case and +0-29m. in the second. They 
appear to be accidental in character, though in some 
cases (notably in Class G5) the residuals for the stars 
of the two sets are similar in sign and magnitude. 
The large negative residuals for Classes K and K5 
in the clusters arise from the fact that in the Hyades, 
which contribute most of these stars, only the brighter ; 
ones have had their proper-motions determined, and 
get into our lists, as is clear from examination of 
Big 2: 
Among the dwarf stars, therefore, a. typical star 
of any spectral class is about seven times fainter than 
one of the preceding class, and seven times brighter 
than one of the following class. 
The giant stars of all the spectral classes appear 
to be of about the same mean brightness, averaging 
a little above absolute magnitude zero, that is, about 
Nigu'9393,) VOR, G3] 
| 
; 
| 
’ 
a hundred times as bright as the sun. Since the 
stars of this series which appear in Fig. 2 have been 
selected by apparent brightness, which gives a strong 
preference to those of the greatest luminosity, the 
average brightness of all the giant stars in a given 
region of space must be less than this, perhaps con- 
siderably so. 
By tabulating the residual differences between the 
absolute magnitudes of the individual dwarf stars and 
the values given by the formulz just described, we 
find that the average difference, regardless of sign, 
for the stars of measured parallax is +0-88m. for 
spectra A to F8, +1-02m. for spectra G and Gs, end 
+1-15m. for K and M. For the stars in clusters, the 
average differences are +0-7om. for spectra BO to 
Bg, +0-66m. for A and As5, +0:56m. for spectra F 
to F8, and +0-80m. for G and G5. 
These differences are larger for the stars of 
measured parallax than for the others (probably on 
account of the greater average uncertainty of the 
individual parallaxes and spectra in this case), but 
show no marked systematic variation with the class 
of spectrum. Their distribution follows very approxi- 
mately the law of accidental errors, as is shown by 
Table VI., in which the observed numbers lying 
between certain limits are compared with those given 
by this law 
TaBieE VI. 
Distribution of Differences from the Typical Absolute 
Magnitudes. 
Stars with measured parallax Stars in clusters 
Limits Observed Theory Limits Observed Theory 
ml. m. m. m. 
+0'0 to +08 65 61 +0'0 to +0°5 59 58 
+0°8 to +1°6 41 44 +o0°5 to +1°0 42 42 
+1°6 to +2°4 21 23 +1°0 to #1°5 21 24 
a22-A tOp==ae2 10 9 +2155, toy 4270 10 8 
+3°2 to £4'0 3 3 =t2'0) towats 245 4 4 
The theoretical distribution for the stars in clusters 
corresponds to a probable error of +0-61m., and that 
for the others to one of +0-94m. Correction for the 
known influence of uncertainties of the parallaxes and 
spectra would reduce the latter to about +0-75m. It 
appears, therefore, that the absolute magnitude of a 
dwarf star can be predicted with surprising accuracy 
from a mere knowledge of its spectrum. Half of all 
the dwarf stars are not more than twice as bright 
or as faint as the typical stars of their spectral classes. 
The corresponding uncertainty in the estimated 
parallax would be about one-third of its amount. 
The parallaxes of the giant stars are so small, in 
comparison with the errors of even the best present 
methods of observation, that direct observations are 
not well adapted to determine to what degree they 
differ in brightness among themselves. An indirect 
method of determining this is, however, practicabie, 
among those classes in which all the naked-eye stars 
are giants, by comparing the parallactic motions of 
those. stars the proper-motions of which at right 
angles to the direction of the parallactic drift are 
larye and small. A discussion by this method of the 
| typical case of Class M (the tetails of which will be 
given elsewhere) shows that, if the distribution of 
the absolute magnitudes of these stars also follows 
the “law of errors,” the probable error correspond- 
ing to it is approximately +0-6m.—almost exactly the 
same as has already been found for the dwarf stars. 
The mean absolute magnitude of all the stars of this 
class which are visible to the naked eye is —o-5, and 
that of all the stars in a given region of space 1s 
406. This method can scarcely: be applied to the 
naked-eye stars of the other spectral classes (unless 
some way can be devised for weeding out the dwarf 
stars from among the giants); but it seems probable 
