10 



c. V. L. Cliarlier 



where .^(0)4^0. Then we get, for d = 0, 



00 



0 



0 



independent of d. 



In otlier words: the mean value of the distance converges with evanescent d 

 ahvays against a finite hmit (though this limit may not necessarily have the value Jf J. 



The practical conclusion of the preceding investigation is that the mean di- 

 stance of the clusters increases inversely as the apparent diameter for large values 

 of the apparent diameter that, however, for small apparent diameters the mean 

 distance approaches to a certain finite limit 



This is exactly what was found in the example discussed above. For having 

 the line of regression between apparent diameter and the distance we ouglit to 

 know the frequency distribution of the clusters in space as well as the distribution 

 of their diameters. It is, for our main purpose, sufficient to know the former 

 frequency distribution (determined by /^(r)) as the constant depends only on the 

 characteristics of this distribution. In practice we may therefore proceed in such 

 a way that we first assume the distance to be inversely proportional to d, for all 

 values oE d, and in this manner deduce an approximate image of the distribution 

 of the objects. From that we may numerically deduce the characteristics of /j(r) 

 and thus get a second approximation to the value of the distance. Regarding the 

 clusters (as well as the globular clusters) I have been obliged to confine me to the 

 first approximation — with some slight modifications — because of the small 

 number of the clusters that for the present can be taken into account. 



5. The position of the clusters has been referred to a galactic system of 

 coordinates. As such I used first the galactic system determined from the B-stars 

 in Meddel , Ser. II, N:o 14. It was, however, found that the clusters determined 

 a galactic plane rather different from the XF-plane of that system. A similar 

 result was obtained from the variables of short period, which were at the same 

 time discussed. I subsequently found that the determination of the XF-plane of 

 the Galaxy from the J5-stars, though the number of stars was rather great, suffered 

 from not inconsiderable mean errors, which can be inferred from the distribution 

 of the i?-stars in the FZ-plane shown on plate IV of the cited memoir. Even the 

 0-stars, discussed by Gyllenbeeg in Medd. N:o 75 furnish evidence in the same 

 direction. I therefore have preferred to accept the galactic plane used by Pickering 

 in all researches made at the Harvard observatory, having first convinced myself 



* Similar conclusions hold true even ref^arding otlier attibutes of the stars as for uistance 

 their apparent magnitudes and their proper motions. 



