The general Characteristics of the frequencyfunction of stellar movements 13 



Putting further 



B m = <V a,/ o a * R'i 



we can demonstrate that 



(22) Y,Z){1 + v ß.., i,-..,). 



1 -n.t> 



Here 



(22*) f = U'x*m+ Y^ + Z^+2YZ^- + 2XZ^- + XY*1 ] 

 y 1 ■ S\ dr xx ^ dr yy T dr zz ^ dr yz ^ dr xz T dr xg j 



if by iS we denote the determinant 



8: 



v, r zy , r 



and it is understood that r xx = r yy = r zz — I. Furthermore, it may be shown that 

 the polynoms Riß as parameters only contain the correlation coefficients r xlJ , r xz , r yz . 

 Then, if as system of coordinates we choose a system where the correlation coefficients 

 vanish, which may always be done by the linear transformation (7) and the cubic (8), 

 the functions f' and Rl jk are independent of any parameter whatsoever and conse- 

 quently may once for all be tabulated. As then the variables may be separated it 

 will be seen that 



■Rijk =z Rioo ■ Roju • Hook , 



and if 



'f!: :) (X) = - fo (X)B ( „„ 



we obtain 



(23) ^ = ? 0 (X) . <p 0 ( Y) . % {Z) + S ß 07 , rf» (X) ( Y) 9 f\Z) . 

 The functions 



<J> t = j/2^<p 0 (X), <l>;+, = l/2w^(«), are tabulated by Bruns* for all i > 5, 

 and, save for i — 1 and i =2 i = h, by Chablier **. 



Thus the frequency of any »point» X, Y, Z may be easily calculated. 



Without tables of the functions <t>, the computations may be performed, 

 remembering that in the system considered 



B 'i o o = — X 



R' 0 „ = X s — 1 



(24) 200 



i? 300 = 3X— X 3 



fi' 40o =3 — 6X 2 + X 4 , 



* Bruns: Wahrscheinlichkeitsrechnung und Kollektivmasslehre. Verl. B. G. Teubner 19U6. 

 ** Charlier: Reaserches into the theory of probability. Medd. från Lunds Obs. Ser. II N. 5. 



