studies in Stellar Statistics 



43 



From these nvimbers we have as value of the mean absolute magnitude of 

 the stars: 



M=-\- 3"'.76 



and the dispersion (a) amounts to 



0=4- 3'".06 



so that the frequency function for M should be: 



(i1/-3.7li)2 



(73) — — e 



We shall have opportunity beneath to derive the form of this function in 

 another way. Taking into account that the stars, for which the parallax is known, 

 are not chosen at random, we may expect that the expression (73) cannot give the 

 true frequency function for M. 



22. According to our working plan we have to deduce the relation between 

 ;r and m from the proper motions of the stars. Suppose pa. to be the proper 

 motion in a and jjg that in S und let 



P = V{pr, cos + l/x^ + if 



then the most direct method should be to compute, for each square, the mean 

 value of p for different magnitudes and hence deduce the value of X with the 

 help of (69). 



Let us examine the frequency curve of p. The simplest assumption regarding 

 the frequency curves of x and y is 1) to accept them to be frequency curves of 

 type A and 2) to regard x and y as independent one of the other. Though the 

 second assumption is no longer possible to maintain — as I hope to have oppor- 

 tunity to nearer discuss in the next term — it is of interest to examine the frequency 

 curve of p resulting from these two assumptions. 



According to our first hypothesis (that is generally nearly fulfiilled) the pro- 

 bability for a value x ± ^/a dx is 



-_ e 2oj^ 



and the corresponding probability for a value y± ^/2dy 



dy 



e 



0,1/27C 



Supposing moreover = = a (a supposition that is, indeed, generally not fulfilled) 

 the probability that simultaneously x and y lie in the said limits is 



,2 ' 



27r 



e 



