be just about on the limit. In this value, 1", is included the effect of the parallactic motion, which, on the general average, increases the apparent proper motion of a star. To study this effect let us call the list of 90 or more stars actually found List A. Were it possible to observe the proper motions of the stars themselves separate from the parallactic motion, we should find that, when we enumerate all having a proper motion of more than 1", we should add some to our List A and take away others. The stars we should add would be those moving in the same direction as the sun, whose motions appear to us to be smaller than they really are, while we should take away those moving in the opposite direction, whose motions appear to us larger than they really are. On the average, we should take away more than we added, thus diminishing slightly the number of stars whose motion exceeds 1". Making every allowance, we may estimate that probably 80 stars have an actual proper motion on the celestial sphere of 1" or more. We have found that the average linear proper motion of a star, as projected on the sphere, is about 6 radii of the earth's orbit annually. A star having this motion would have to be placed at the distance 6R to have, as seen by us, an angular motion of 1". The parallax corresponding to the surface of this sphere is 0".167. The volume of the sphere is 216, and according to our estimate from the parallaxes it would contain only 27 stars. It will be seen that these results give a greater density of the stars than the result from the measured parallaxes; that is to say, they indicate that there are still an important number of measurable parallaxes to be determined, while the number of stars is less than would be derived from their proper motions. But the fact is that the number of stars estimated as within a given sphere by the proper motions will be in excess, owing to the actual diversity of these proper motions, which may range from to a value several times greater than the average. In consequence of this, our list of stars with a proper motion exceeding 1" will contain a number lying outside the sphere 6R, but having a proper motion larger than the average. We are also to consider that within the sphere may actually lie stars having a proper motion less than the average, which will, therefore, be omitted from the list. Of the number of omitted and added stars the latter will be the greater, because the volumes of spheres increase as the cubes of their radii. For example, the space between the spheres 6R and 9R is more than double that within 6R, and our list will include many stars in this space. The discrepancy between the parallaxes and the proper motions probably arises in this way.
Let us see what the result is when we take stars of smaller proper motion. The most definite limit which we can set is 10" per century. We have seen that Dr. Auwers, in his zone, found 23.9
