34 



Walter Gyllenberg 



(31) 



where 



\ 



A = 



N 



- L * 0 2 0 



N 









B = 



^002 



N 



XT 2 0 0 



- 





A, 





^0» 



N 



-"020 









Z) = 



^no 







N 



-"200 



\ 



/<; = 



^.M 



^no 





N 



-"020 



à 1 



F = 



^,0, 







N 



•"00 2 





A, = 



^2 0 0. 



- tr 1 1 0' 





(32) 

















y 0II , 



N 



The relations (30) and (31) exist in fact between the coefficients and the moments 

 when referred to any system of coordinates. It is further easily shown that the 

 determinants in the formulae (30) and (31) are connected through the relation 



A = 1 : A,. 



31. Using exclusively observations in the line of sight, the third relation in 

 (30), coincident with (29), gives us the equation of condition. Solving the unknown 

 quantities in the right member of (29) by the method of least squares we get in the 

 usual way the squares of the three axes of the velocity ellipsoid expressed through 



(33) 



where 

 (34) 



are the roots of the equation 



; 3 — {A" + B" + C") s 2 + {A"B" + B"C" + C'A" — D" 2 —E" 2 - F" 2 ) s — A = 0. 

 The direction cosines X, fj. and v of the three axes are finally given through 



(35) 



{A" — Si )l+ F"\l f £"v = 0, 

 F"\ 4- (B" — Si) fj. + D"v = 0, 

 E"\ + D "[x 4- (C" — Si) v = 0, 



where for a certain axis the corresponding value of Si is to be substituted. 



32. There are twenty-four equations of condition of the form (29) to be solved. 

 The number of stars in the different spectral classes being rather small for separate 

 solutions, I have found it convenient to treat them in the three following groups: 



Group Spectral classes Number of stars 



1 B and A 510 



2 F and G 445 



3 K and M 571 



