CHAPTERS ON TEE STARS. 463 



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 cor- 

 responding 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 par- 

 allaxes; that is to say, they indicate that there are still an important 

 number of measurable parallaxes to be determined, while the num- 

 ber 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 6E and 9R is more than double that within 

 6R, and our list will include many stars in this space. The discrep- 

 ancy 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 cen- 

 tury. We have seen that Dr. Auwers, in his zone, found 23.9 



