studies in Stellar Statistics 



45 



Unfortunately our assumption that x and y are independent variables is not 

 fulriUed (not even does Oj = o,) There is, however, a rather pronounced correlation 

 between these variables and taking that into consideration the frequency function 

 for p is much more comphcated than (74**). We shall in like manner find that 

 in most cases met with in statistics the fi'equency function of polar coordinates is 

 less simple than that of the reclilinear coordinates For these reasons the use of 

 p in our present problem is to be avoided *). The coordinates x and y themselves 

 are indeed better adapted to the solution of our problem. 



23. We have 



(75) 

 and 



- \y, 



— X, hn 



M4p) = E,e =K,e 



ilf^TT) = K, e- "'"^ = K, e- 



where h 



0.2 



mod 



\, = o[\og M,n,(p)- log M,4p]] 



(X = X, b). 



We have to determine X, or better X^, from the observed proper motions in 

 a and S. From (75) we deduce, taking into account the value of h, the following 

 formula for Xj : 



(75*^ -^^ _ 5[log M ,n,{p) - log M,4p)] 



or 



(75**) 



supposing m.^ 



This form.u]a should be used, if we had deduced the mean value of p (of the 

 preceeding lecture), which coincides with M{p). If, however, the coordinates x and 

 y are used we could substitute in (75**) for 31(2^) the mean value — M{x) — of 

 X or that of y. A better result is sometimes obtained if the dispersions in x and 

 y — which are evidently inversely proportional to the distance — are introduced 

 into (75*). Our formula then is: 



(75***) Xj = ^ _ log ^ 



-nij — 1. 



111, 



where corresponds to the magnitude m, , to the magnitude . 



As an illustration I give the correlation table for the square 63 between 

 X (= Aa cos §) and y {= AS) for the 5th magnitude : 



Correlation table for C3. 



Class 



- 1 



0 



+ 1 



+ 2 



+ 3 



+ 4 



+ 5 



+ 6 



+ 7 



+ 8 



+ 9 



+ 10 



Sum 



— 1 





2 



6 





2 

















10 



0 



1 



14 



36 



7 



12 







1 











71 



+ 1 



1 



7 



4 





1 















(1) 



(14) 



Sum 



2 



23 



46 



7 



15 





1 







1 (1) 



(95) 



*) Log p is better suited for discussion. 



