24 



C. V. L. Charlier 



the table of the probabüity-function calculated by Sheppard (Biometrica II). In 

 this manner we have the table 







3^ 



Dir); c- 



D{r) 



1 



0 



— 0 328 



0.378 0 



1.010 



10 



— 5 



- \.n 



0.189 5 



0.501 



J 00 



— 10 



— 2.12 



0.042 2 



0.112 



inoo 



— 15 



— 3.01 



0.004 30 



0.011 4 



10 000 



- 20 



— 3.91 



0.000 191 



0.00>) .505 



100 000 



-25 



— 4 to 



0.000 003 96 



0 000 010 5 



1 000(100 



- 30 



— 5.70 



0.000 000 035 1 



0.000 000 093 



10 000 000 



— 85 









11. The metliod of Kapteyn, original and important though it is, has, 

 methodically, a serious defect that may indeed be caused by the imperfect state of 

 the statistical material, but tliat must necessarily have a systematic influence on the 

 formula obtained for the number of stars of a given magnitude. His above results 

 are namely deduced from observations all over the sky. Suppose, however, for a 

 moment the Milky Way to have the form of a flattened cylinder and assume the 

 stars — all of the same luminosity — to be uniformally distributed in this cylinder, 

 then the number of star having an apparent brightness greater than, say, h should, 



for great values of h, vary as h'~^'' and for small values as h~^. Using our for- 

 mula on these observations we should evidently get as result a density of the Milky 

 Way decreasing as the distance from the sun is increasing, though, according to 

 our assumptions, the density is throughout the same. 



This systematic error may partly be avoided, if stars of the same galactic 

 latitude are considered together. This is indeed the line of research followed by 

 Seeligek and Kapteyn in most of their investigations. Taking into account the 

 very irregular form of the Milky Way even this procedure must be regarded only 

 as a necessary assistance. The only rigorous method of treating the problems of 

 stellar statistics is to take out a well defined definite region of the sky and to 

 study the stars in this region with respect to brightness, proper motions, distances 

 of double stars and other character. If a single region of the sky, however 

 small it may be, should be thoroughly examined for all stars, say, to the 20.-th 

 magnitude, we should know more of the construction of the heavens than we now 

 can come to know from all accessible observations made all over the sky. 



Uufortunately no region of the sky is so thoroughly examined. The material 

 of observations existing may however be examined for different parts of the sky. 

 For effecting such an examination I divide the sky into 48 parts all of the same 

 area (= 860 in the way shown by the accompanying diagram. For each of 

 tliese squares (I shall call each of these 48 parts a » square») I have collected, as 

 far as possible, all available informations regarding brightness, proper motions, 



