May 21, 1914| 
NATURE 
297 
of stars in the form of an oblate spheroid, equa- 
torially surrounded by a belt of irregular star- 
clouds composing the Milky Way. In connection 
with such a theory it is of obvious importance to 
know whether the condensation towards the galaxy 
shown by the stars of magnitudes five to nine 
persists for still fainter stars, and in what degree. 
Very different views have been held on this matter. 
In regard to the stars classified according to their 
visual magnitudes, Kapteyn concluded that the 
galactic condensation increases very much with 
diminishing brightness, giving its value for all 
the stars brighter than 17” as 45. Pickering, on 
the contrary, found no marked change in the 
relative densities in the galaxy and at its poles, 
down to the thirteenth magnitude (the limit of his 
data). The counts on the Franklin-Adams plates, 
based on a photographic magnitude classification, 
lead to a similar result ; the density of all the stars 
brighter than 17” in the galaxy does not exceed 
six times that at the galactic poles, and the ratio 
is perhaps not more than four. Although it is 
possible that there may be a systematic change 
of colour of the stars with increasing faintness, 
different in different galactic latitudes, which 
would make results derived from counts of stars 
of determined visual magnitudes differ systematic- 
- ally from those based on photographic magnitudes, 
yet such evidence upon the point as already exists 
renders it probable that the galactic condensation 
is in either case nearly constant from the sixth 
to the seventeenth magnitude. 
On this account the rate of increase in the 
number of stars per magnitude will be nearly 
constant all over the sky, so that this rate may 
conveniently be studied from a table giving the 
numbers of stars in the whole sky brighter than 
each magnitude m; this will be denoted by Nn. 
All the best available data have been embodied 
in the following table, giving N,» for values of m 
down to 17. The values of An(=log Nmn+1/Nm) 
are also given, as they provide a measure of the 
geometric ratio of increase in the number of the 
stars. 
Taste I.—The Number of Stars in the Whole Sky 
Brighter than Magnitude m. 
mt Nom log Nm Am 
2 38. © is. Bene 
3 ETI. ‘sen Seem 0-47 
4 300 2-48 0-43 
5 950 2-98 0-50 
6 3,150 3°50 0°52 
7 9,810 3°99 0-49 
8 32,360 4°51 0°52 
9 97,400 4:99 0-48 
10 271,800 5°43 0-44 
II 698,000 5°84 O-41 
I2 1,659,000 6-22 0:38 
13 3,682,000 6:57 0:35 
14 7,646,000 6-88 0-31 
ors 15,470,000 7-19 0-31 
16 29,510,000 TAT 0:28 
17 54,900,000 774 0°27 
The data for the stars of magnitudes 2 to 6 
are somewhat uncertain, which accounts for the 
irregular run of the first few values of An, but 
NOh 2925.5 VOL..193)| 
beyond this point the steady decrease in A, is 
very noticeable. This clearly shows that modern 
photographic telescopes now penetrate to regions 
of space where the stars begin to thin out in 
numbers to a quite considerable extent, for it is 
easy to prove that if the stars were distributed 
uniformly throughout space, A,, should preserve 
the constant value o'6. This assumes, what 
appears to be fairly correct, that any possible 
absorption of light in space does not materially 
diminish A,. 
Irom the numbers in the foregoing table, the 
following simple rational formula can be derived, 
aN ‘ 
log —"=a+bm—cm?, 
am 
or, in an equivalent form, 
Nea! Pe "Biw2 — oo) 
iat 
The latter formula, which is the integral of the 
error curve, implies that the total number of the 
stars is finite, and this is now generally accepted 
as true; A represents this total number, while C 
denotes the magnitude which divides all the stars 
into two equal groups, those brighter being equal 
in number to those fainter. A, B, C can be 
deduced from a, b, c, which are readily obtained 
from the observed values of Nm, but A is not 
narrowly determined—its value seems to be not 
less than tooo million, and probably not greater 
than 2000 million, so that the total number of the 
stars 1s comparable with the population of the 
earth (this is roughly estimated as 1600 million). 
The constant C is more closely determined, and 
is approximately 23 or 24. Stars of this magni- 
tude could just be photographed, with many hours 
exposure, with the largest telescope in the world, 
the 60-in. reflector at the Mount Wilson observa- 
tory. There remain, therefore, beyond our present 
powers of exploration still fainter stars equal in 
number to all those which could possibly be 
examined at the present time. 
These impressive numbers shrink into a smaller 
compass when the total light of the stars is con- 
sidered. It may readily be shown that if the 
formula for Nm is correct, the total intensity I, 
of all the stars brighter than magnitude m can be 
represented by an expression identical in form 
with that for Nm; but whereas the peak for the 
error curve, the integral of which represents N,, 
or In, is in the former case (C=23 or 24) beyond 
the limits of the observed data, in the case of Im 
it is well within these limits—in fact, half the 
total light of the stars comes from those brighter 
than about 9°5”. Up to this point the light re- 
ceived from all the stars of magnitude m to m+1 
increases; beyond this it diminishes rapidly, the 
increase in the number of the faint stars, great 
though it is, being insufficient to counterbalance 
their diminished brightness. Owing to the formula 
for Nm giving too small a number of bright stars 
(a defect of little moment for most values of m, in 
the case of Nn, but of serious importance when the 
