III. 



Moments of higher order. Hypothesis of two star-streams. 



27. The object of this investigation being to give an empirical description of 

 the frequency distribution of the velocities of the stars, independent of any theore- 

 tical considerations regarding the motion of a system of bodies like our Milky Way, 

 the most general assumption to be discussed is that formulated in formula (51*) of 

 § 18. The first term in this formula gives what is usually called the ellipsoidal hy- 

 pothesis, or, in the language of mathematical statistics, the normal correlation func- 

 tion of the velocities. I have discussed in the preceding section the characteristics 

 of this normal function. The third and the higher dérivâtes of this function 

 multiplied by properly determined coefficients give the corrections for obtaining the 

 general form of the correlation function. 



The coefficients of the derived functions of the tJiird order, eventually multi- 

 plied by certain numerical factors, are called the coefficients of the shewness. The 

 coefficients of the derived functions of the fourth order, also multiplied by certain 

 factors, are called the coefficients of the excess. For our present purpose it is suf- 

 ficient to calculate four of these coefficients, namely Su, Sv, Eu, Ev, defined in the 

 formulée (37) and (38). 



The value of these coefficients have been given in tab. V. 



We conclude from this table that the excess in U {Eu), as well as the excess 

 in V [Ev), is decidedly negative (unlike the excess of the observed proper motions 

 which was as decidedly positive). The mean value in both coordinates is — 0.206 and 

 far too large to be explained by accidental errors. 



As to the shewness we may observe the systematic variation in the (Sjj-coeffi- 

 cient, especially in the C- and the Z)-squares. 



The existence of appreciable values of the coefficients of the skewness and of 

 the excess must from these reasons be considered as established. 



Hence the conclusion: The terms of the higher orders in [51*) give sensible con- 

 tributions to the frequency function of the velocities of the stars; or in other words: the 

 ellipsoidal hypothesis does not suffice for explaining the observed proper motions of 

 the stars. 



