The general Characteristics of the frequencyf unction of stellar movements 71 



x ) 2 00 



—- 3.07 q 



— 0.37 



~ \fl )o2U 



= 4 .24 q 



— 0.98 



y = (H\ 02 



= 1.40 q' 



— 0.18 



The rigorous solution is 



q = 0.B2 X = 1 .23 y = 0.36 



which gives especially an absurdedly low value of y — (H 2 ) Q0r From equations (85) 

 we now find for H and Jc the equations 



1 .23 = jfe*(0.61 H 2 4- 0.39) 

 0.36 = F(0.19 H 2 4- 0.81), 



which give an imaginary value of H. This is in itself nothing surprising, it only 

 means that with such values of x and y as above it is an absurdity to assume one 

 value of ftj to be common for galactic latitude less than 30° and another value to 

 be common for the rest of the sky. But as before said the rigorous solution of 

 the equations (86) is illusory. Taking q =0.75 we find 



(H\ 00 = 1.93 (2P) 020 = 2.10 (iP) 002 = 0.87 



by which it is seen that here also we may regard (IP). 2W and (H 2 ) 02(j as equal, a 

 thing that accentuates the fact that a solution for q here is quite unsharp, even if 

 at any price we should require the ellipsoids of the cross-motion and radial motion 

 to coincide. Evidently the assumption on which the equation (85) is based is still 

 insufficient with the above values of the {H 2 )^, but we may conclude that in order 

 to make the dispersions of stellar motion along the axes of sysfem III compatible with 

 the values found from the radial velocities we should have to assume that the mean 

 parallax of the stars brighter than 6.0 in the galactic polar regions is about double 

 the mean parallax of the stars in the Galaxy. 



Evidently the assumption of a varying d- 1 does not suffice to explain the 

 disagreement between the two ellipsoids. 



As regards the higher characteristics, it is principally the ß-characteristics 

 that interest us. We remind of the formula 



SW = - 



r -" 200 1, 02» x, 002 



and see by the equation (85) and the table XV that they will be nearly independent 

 of the variation in & 1 at least when H is as small as 1.2. Accordingly we will not 

 trouble for their corrections; it will suffice to hold in mind that the coefficients 

 of skewness are found slightly too small and the coefficients of excess slightly 

 too large. 



