20 PRESIDENT’S ADDRESS. 
these true parallaxes with the mean parallax of stars having correspond- 
ing magnitude and proper motion, and this comparison will lead to a 
knowledge of the frequency-law required. It is true that, owing to the 
scarcity of material at present available, the determination of the 
frequency-law is not so strong as may be desirable, but further improve- 
ment is simply a question of time and the augmentation of parallax- 
determination. 
Adopting provisionally the frequency-law found in this way by 
Kapteyn,' we can localise all the stars in space down to about the 
ninth magnitude. 
Take, for example, the stars of magnitude 5:5 to 6:5. There are 
about 4,800 of these stars in the whole sky. According to Auwers- 
Bradley, about 9} per cent. of these stars, or some 460 in all, have 
proper motions between 0-04 and 005. Now, according to Kapteyn’s 
empiric formula, whose satisfactory agreement with the Yale results has 
just been shown, the mean parallax of such stars is almost exactly 0-01. 
Further, according to his frequency-law, 29 per cent. of the stars have 
parallaxes between the mean value and double the mean value; 6 per 
cent. have parallaxes between twice and three times the mean value ; 
14 per cent. between three and four times the mean value. Therefore 
of our 460 stars 133 will have parallaxes between 0’-01 and 0’-02, twenty- 
eight between 0-02 and 0/03, seven between 0-03 and 0/04, and 
so on. 
Localising in the same way the stars of the sixth magnitude having 
other proper motions, and then treating the stars of the first magnitude, 
second magnitude, third magnitude, and so on to the ninth magnitude in 
the same way, we finally locate all these stars in space.” 
It is true we have not localised the individual stars, but we know 
approximately and within certain limits of magnitude the number of stars 
at each distance from the Sun. 
Thus the apparent brightness and the distance being known we have 
the means of determining the light-energy or absolute /wminosity of the 
stars, provided it can be assumed that light does not suffer any extinction 
in its passage through interstellar space. 
On this assumption Kapteyn was led to the following results, viz., 
that within a sphere the radius of which is 560 light-years (a distance 
which corresponds with that of the average star of the ninth magnitude) 
there will be found :— 
1 star giving frora 100,000 to 10,000 times the light of our Sun. 
26 stars 4 10,000 ,, 1,000 a a a 
130005 + L000) 155 -beelOO™ iia, ” ” 
22,000 ,, + 100 —,, 10 » ” 59 
140,000 ” ” 10 ” 1 ” ” ” 
430,000 ” ” 1 ” 01 ” ” ” 
650,000 ” ” 01 ” 0-01 ” ” ” 
1 Publications Astron. Lab. Groningen, No. 8, p. 23. 
2 Tbid., No. 11, Table II. 
