456 POPULAR SCIENCE MONTHLY. 



than from the more distant stars was supposed to lead to the 

 conclusion that the latter had a more rapid real motion than the 

 former. A little thought will show that, while this is quite true of the 

 stars included in the list, this does not prove it to be true for the stars 

 in general. We can not, as already pointed out, determine the motion 

 of any star unless it exceeds a certain limit. Hence, in the case of the 

 more distant stars we can observe the proper motions only of those 

 which move most rapidly, while in the case of the nearer ones we may 

 have measured them all. We should, therefore, naturally expect that 

 the more distant stars in our list will show too large a value of the 

 proper motion, for the simple reason that those having small proper 

 motion are not included in the average. There is, therefore, no evi- 

 dence that the more distant stars move faster than the nearer ones. 



An error in the opposite direction occurs through the method of 

 selecting stars of given proper motion. We have already pointed out 

 that in the case of any individual star we cannot determine how much 

 of its apparent apical motion may be that of the star itself, and how 

 much the parallactic motion arising from the motion of the earth. 

 What we have done is to assume that in the case of a great number of 

 stars the actual apical motions will be equal, and in the opposite direc- 

 tions, so as to cancel each other in the average of a great number, leav- 

 ing this average as the parallactic motion. Now, to fix the ideas, 

 suppose that two stars have an equal apical motion, say 3 radii of the 

 earth's orbit in a year, but in opposite directions. The apical motion 

 of the earth being 4 radii per year, it follows that the star which is 

 moving in the same direction as the earth will have a relative apical 

 motion of only 1, and will, therefore, not appear in our list as a star 

 of large proper motion. On the other hand, the star moving with equal 

 speed in the opposite direction will have a motion of 7 radii per year, 

 and, will, therefore, be included among stars of considerable proper mo- 

 tion. Thus, a bias occurs, in consequence of which we include many 

 stars having a motion away from the solar apex, while the correspond- 

 ing ones, necessary to cancel that motion, will be left out of the count. 

 Thus, the parallactic motion will, in the average, be too large in the case 

 of the stars of large apparent proper motion. Now, this is exactly 

 what we see in the above tables. As we take the classes with larger 

 and larger proper motions, the supposed parallactic motion, which is 

 really the mean of the apical motions, seems to increase in a yet larger 

 degree. It is, therefore, impossible to determine from comparisons like 

 these what the exact ratio is. 



This error is avoided when we do not arrange and select the stars 

 according to the magnitude of their proper motions, but take a large 

 list of stars, determine their proper motions as best we can and draw 

 our conclusions from the whole mass. This has been done by Kapteyn 



