stellar clusters J 



Hence we find that the mean distance of clusters having a vanishing apparent 

 diameter does not, for b = 0, converge against an infinite value as would be the 

 case if the mean distance were inversely proportional to the diameter. 



Let further h be very large. We then exchange, in the integral, r for a new 

 variable 1/ = rb and get 



{r) = I ['f 'hiî/) fdll : I'f ,(f ) Ul^ y dy . 



If now b is very large, the function f^(y/b) converges against 'fi(O), which 

 according to our assumptions is finite. Consec|uently we get for large values of b 



(5) ■ M„{r)^^, 



where 



(5*) "^"^^^ '^'^'^ ■ J "^^^''^ ^ ■ 



0 0 



We thus arrive at the conclusion that the mean value of the distance of a 

 cluster may be deduced from its apparent diameter with tlie help of the formula (5) 

 as soon as b is large, tvhereas this formtda is no longer applicable for vanishing values 

 of the diameter. 



The value of and may be expressed into the characteristics of the 

 function tp^ and w^. The constants If, and K,^ are, indeed, directly expressed 

 through the moments of the functions rp, and respectively and, according to 

 known formulae we hence have 



(6) 



where m^ and m^ denote the means, and the dispersions belonging to the 

 functions 9^ and . 



The results above are deduced under the supposition that tp^ and (p^ are both 

 frequency functions of the type B. As to (p^ this supposition is certainly fulfilled, 

 as can be found from the repartition of the clusters discussed below. Regarding 

 ^2 we are for the present in the incertainty. We know that tp^ only exists for 

 positive values of B but it may be that cp2(0) — 0. Even in this case our con- 

 clusions are however (under certain conditions) valid. Suppose indeed that, in the 

 neighbourhood of z==0, 



(p2(£c) = x^'g[x) , 



2 



