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 evidence 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 directions, so as to cancel each other in the average of a great number, leaving 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 motion. Thus, a bias occurs, in consequence of which we include many stars having a motion away from the solar apex, while the corresponding 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
