AvcusT 6, 1914| 
follows that these stars are, on the average, from two 
to eight times as bright as the sun. The A-F stars 
are a little, but not much, further away, the stars 
fainter than 9°5 magnitude being at an average 
distance of 263 parsecs. At this distance the sun 
would have a magnitude of 125, and these stars are 
from sixteen to four times as luminous as the sun. 
It has been shown how the knowledge that the 
solar system is moving in a known direction with a 
velocity of 19:5 km. per second leads to a determina- 
tion ot the distances of groups of stars the angular 
movements of which are known. The hypothesis 
made is that in a number like one hundred or two 
hundred stars, the irregular angular movements due 
to the motions of the stars themselves neutralise one 
another on the average. But this is only the mean 
distance of the group, and some are much nearer 
and some much further. ‘the distribution of the stars 
about this mean distance may be derived from the 
proper motions, if we know how the linear velocities 
are distributed. I shall apply this method to the 
group of stars which are like the sun in type of 
spectrum, and therefore, presumably, of like tempera- 
ture and physical constitution. 
Dividing these into three classes according to their 
magnitude, it is found that their parallactic motion 
due to the sun’s movement, and their average motion 
in the perpendicular direction due to their own peculiar 
movements, are as follows :— 
Parallactic Ay. cross 
No. motion motion Ratio 
All stars down to IIl’om. 1247 192 +1°67 0 87 
Stars brighterthanto‘7om. 470 2°50 +210 084 
= 5 Qyom. . Ween sos eee'Oo 0°87 
In the last column is given the ratio of the average 
cross motion to the parallactic motion. The agree- 
ment of the numbers shows that the bright stars and 
the faint stars have the same average velocity. 
Taking the velocity of the sun as 195 km. a second, 
it follows that the average velocity of these stars in 
the direction perpendicular to the sun’s motion is 
13°7 km. a second. 
We shall now make the assumption that some of 
these stars are moving faster than this velocity and 
some slower, just as errors of observation are dis- 
tributed about a mean error. With a mean velocity of 
13°7 km. a second, there will be in 1000 stars 
231 with velocities oto 5 km/sec. 
208 ” ” 5 » 10 
175 ” ” TO ,, 15 
141 is = LS 920 
163 - - 20 5, 30 
59 ” ” 39 5, 40 
18 ” ” 40 ” 50 
I ” ” 750 
If now the observed proper motions are arranged, 
it is found that the number less than any value 7 
can be represented satisfactorily by an algebraic 
/ 
(7? +a?)? 
stars and a is the mean value of 7. The following 
table shows the actual number of stars with proper 
motions between certain limits, compared with the 
number given by the formula :— 
formula N where N is the total number of 
Tasie VII. 
Limits of ay motion Ne. eee eee Difference 
o” to 1” a century 427 "429 —2 
Leen es9 346 337 +9 
2) eae 53 324 332 —8 
4 5 7 » 105 103 +2 
7g LO e ep. 2(- 22 +3 
> 10 r “s 20 19 ar! 
NO. 2336, VOL. 93] 
NATURE 
601 
We may take it that the formula substantially 
represents the observed facts. With the proper 
motions distributed according to this formula, and the 
actual velocities distributed according to the law of 
errors, the distribution of the stars in distance can be 
determined, and it is found that these 1247 stars are 
distributed in space as shown in Table VII. 
TasL_e VIII.—Number of Solar Stars (Types F and 
G) at Different Distances. 
Distabpe Out of total disee HOCRee dentate 
(parsecs) He YELM than 10’0om, than g’om. 
< 100 a I2I 76 40 
100-—200 oes 298 161 65 
200—300 disc 332 136 34 
300— 400 eer 254 68 8 
400 —500 re 146 ae 23 soe I 
500—600 eas 65 ae 5 
600—700 aa 23 
>700 Aas 5 
The most remarkable feature of this table is that 
70 per cent. of the stars lie between the narrow limits 
of one hundred and four hundred parsecs. 
I have treated the 470 stars which are brighter than 
10.0 magnitude and the 148 brighter than 9’0 magni- 
tude in a similar manner. The results are given 
in the third and fourth columns of Table VIII. 
Taking the differences, the distribution in distance 
of the 777 stars of magnitude Io’o-11’0 and of the 
322 stars of g0—I0’0 magnitude is found. 
To compare the intrinsic magnitudes of the stars 
it is convenient to take limits of distance in geo- 
metrical progression with a common ratio 1259 
(log =0;1)," 2.22.40, 50,. 63; /79,, 100, 120, /EEC., pagsece. 
These limits correspond to a change of half a magni- 
tude in the intrinsic brightness of the stars which 
are of the same apparent brightness. Confining our 
attention to the stars of apparent magnitude Io’o to 
II-o, or, speaking broadly, stars of 10-5 magnitude, the 
limits 50-63 parsecs contain stars half a magnitude 
brighter, and distributed over twice the volume of 
those contained between the limits 40-50 parsecs. 
If we may assume that the actual density of the stars 
is the same in all parts of the space with which we 
are dealing, we obtain by reasoning of this kind the 
number of stars between different limits of absolute 
brightness. The following table shows the number 
of stars of different luminosities in a sphere of one 
hundred parsecs radius :— 
Luminosity No. of stars 
(=F 10°0m.—11‘'0om g'om —10’0om 
o'40 to I'o 16,000 18,000 
TeOlgr so 6255 9,500 11,200 
25» 63 5,750 7,300 
635, to 2,570 3,600 
16 5 40 on 502 1,040 
Brighter than 4o... 14 68 
The results in the second column have been 
obtained by considering the faintest stars, those from 
1o’o to 11.0 magnitude. If the class brighter is taken, 
those stars which appear to be of magnitudes 9’0 to 
100, we find in a similar way the quantities given 
in the last column. 
There is an increasing divergence between the 
results. Now it is to be remembered that these 
figures have been derived from regions at different 
distances from the sun. Thus the stars which are 
between sixteen and forty times the brightness of 
| the sun, and which are apparently of magnitude 10 
| to 11, lie between 398 and 631 parsecs, while those 
which are apparently of 9’0 to ro’‘o magnitude lie 
between 251 and 398 parsecs. , 
We may conclude, therefore, that the density of 
