6 



Sven Wickseil 



The assumptions regarding the density function and the luminosity curve of the 

 stars which are then necessary need not be of any high standard of perfection. It 

 is enough if they are fit as functions of interpolation. Fortunately the modern 

 results of stellar statistics give quite enough for this purpose. 



The first to attack the problem of reducing the attributes of the apparent cross- 

 motions to those of the linear motions was Charlier '. For the functions of density 

 and luminosity Charlier made use of forms found in the most modern works 

 of authors such as Kapteyn, Seeliger, Schwartzschild and himself. He assumes 

 the logarithm of the distances of the stars and the absolute magnitudes to be di- 

 stributed as normal Gaussian frequency curves. Then it is shown that the cha- 

 racteristics (moments) of the linear velocities may be expressed in terms of the 

 characteristics of the apparent velocities by means of only two constants. One of 

 those constants is the mean parallax and it may be determined from the mean 

 parallactic motion by comparing with the value of that motion found from the 

 radial velocities. The chief difficulty arises when fixing the value of the other 

 constant, which depends on the variation of the mean parallax for different apparent 

 magnitudes. Having adopted a value of the constant, Charlier investigated 

 the Boss' proper motions assuming as surface of equal frequency an ellipsoid of 

 revolution. 



It is now the aim of this work to seek for a more general characterisation 

 of the distribution. The only assumption made here is that the frequency function 

 is of the general statistical A-type, an assumption of ample elasticity. The charac- 

 teristics are then computed to the fourth order. As the value of the fundamental 

 constant adopted by Charlier is probably too low, and as it is difficult to fix any 

 other definitive value otherwise than from the attributes of the motions themselves, 

 I used a method by which I was able to leave the constants numerically 

 indetermined, thus obtaining the final results explicitely expressed in terms of the 

 constants. It then occurred to me that the fundamental constant might be obtained 

 by comparing the results with those obtained from the radial velocities, of 

 which an investigation has been simultaneously made by my colleague K. A. W. 

 Gyllenberg. As matters have turned out I have come to a modified opinion of 

 the advisability of directly comparing the attributes of the cross-motions and the 

 radial motions, but certain very interesting conclusions may still be drawn. However, 

 other means of determining the constant with a high degree of probability offered 

 themselves. It appeared that nearly all the higher coefficients change sign when 

 varying the constant within a rather narrow range. As those coefficients are 

 coefficients of disturbance of the frequency function from its normal form this fact is 

 of high interest, and makes it probable that the constant has a value within this range. 



As before said the results depend only on the values of two statistical con- 

 tants. The one is the mean parallax and is denoted by and the other is 



1 Charlier, Studies in Stellar statistics II. Medd, från Lunds Observatorium ser. II Nr 9. 



