60 



G. V. L. Chaiiier 



fact was the point of issue of Kapteyn's theory of the two star-streams. Ou i)late 

 I the directious of the hue of symmetry for the different squares are graphically 

 exhibited. 



6. Another character of the correlation tables is of no less importance for the 

 question of the motion of the stars. We find, indeed, that the distribution of the 

 frequencies in the correlation tables is very far from that belonging to a normal 

 frequency distribution. We find this perhaps most easily through examining the 

 frequency curves for the proper motions in right ascension and declination given 

 in the last row and the last column of each correlation table. The skew character 

 of these frequency curves is very clearly pronounced and, what is still more salient, 

 all frequency curves have a very large excess. Certainly, the number of observations 

 in each square is very small, and the mean errors of the higher characteristics 

 (skewness and excess) are considerable. Especially is that the case with the skew- 

 ness. For a nearly normal frequency curve the mean error in the slcewness amounts 

 to about 



± 0.193, 



when the number of observations is 100, and the mean error of the excess to 



± 0.061. 



There is, however, no possibility to explain the large values of S and E 

 as accidental values belonging to a normal curve. It suffices to take account of 

 the mean errors above, but still more conclusive is the systematic character of the 

 values obtained for different squares. In the slcewness this systematic character is 

 more difficult to observe, but regarding the excess we find that this characteristic 

 in all squares (with a few insignificant exceptions) has a decidedly positive value. 

 Tn other words: there are too many stars with very small proper motions. 



One might be inclined to conclude that the frequency curve of the velocities 

 is not of type A but of type B, or, in other words, that the law of Maxwell (even 

 generalized) does not give an approximation to the law governing the motions of 

 the stars. This is, indeed, the conclusion drawn by some writers on this subject. 



There is, however, another explication of this anomaly, which seems to present 

 itself more naturally. 



The frequency law observed is that governing the apparent proper motions of 

 the stars. It is, however, by no moans necessary, neither probable, that the fre- 

 quency distribution of the linear velocities of the stars be proportional to that of 

 the apparent proper motions. Let, indeed, 



'iAU. V) 



denote the frequency function of the linear velocities in a plane perpendicular to 

 the line of sight, and let, as in I, 'f(,{M) be the frequency function of the absolute 

 magnitudes of the stars, then 



Dcü dr X tc JM) clM X cp^ ( U, V) d Ud V 



