44 C. V. L. Charlier 



We here introduce polar coordinates, putting 



X = p cos f, Î/ = P sin f 



and observe that for dx dy we have to put 



dx dy = d[j d'f 

 we get the frequency function 



ix ciX 



dy dy 



d[j d^f p 



or integrating between tp = 0 and (r = 2u 

 (74**) 



which is hence the frequency curve of p. Evidently this curve is of type B, and, 

 regarding the difficulty of obtaining the general form of the frequency curve of 

 this type for continuous variables, it is of interest to discuss somehwat closer the 

 results obtained. 



We observe that 



jdpFip)^^!. 



0 



Let N be the total number of stars and ^(p) the number of stars having a proper 

 motion smaller than p we have 



(74*) ^(p) = [ dp F{p) i\r(i e- fc) , . 



(so that A{co ) = TV). The mean value — J^(p) — of p is 



CX) 00 00 



M{p] 



p dp F{p) : dp F(p) 



0 0 0 



The value of this integral is well known from the theory of probability and 

 we have 



(74) 



M{p) 



Writing 



(74) 0=1/ ^M(p) = 0,797816 M{p) 



this formula can be used for calculating a. Then J.(p) can be calculated from (74*). 



