stars per 100 square degrees having a proper motion of 10" or more. This ratio would give about 10,000 for the whole heavens. The sphere corresponding to this limit of proper motion is 60R. On our hypothesis as to star density this sphere would contain 27,000 stars, nearly three times the number derived from Auwers's work. But it is not at all unlikely that even this sphere in question contains twice as many stars as have been detected. Great numbers of the more distant stars will not have been catalogued, owing to their faintness, because a star at the distance 60R will shine to us with only one per cent, the light of one at distance 6R. This corresponds to a diminution of five magnitudes; that is to say, a star of the sixth magnitude at distance 6R would only be of the eleventh magnitude at distance 60R, and would, therefore, not be catalogued at all. There is, therefore, no reason for changing our estimate of star density, which assigns to each star around us 8 units of volume in space.
This fact suggests another important one. Owing to the great diversity in the absolute magnitude of the stars, those we can observe with our telescopes will naturally be more crowded in the neighborhood of our system than they will at greater distances.
Some further results as to the mean parallax of the stars may be derived from a continuation of the statistical study of the proper motions. Kapteyn's investigation in this direction includes a determination of the mean parallactic motion of the stars of each magnitude for the first and second spectral types separately. From this he obtains the following mean parallaxes for stars of the different magnitudes:
Mean parallaxes of stars of different magnitudes, and of the two principal types, as found from their parallactic motions:
| Mag. | Type I. | Type II. | |||
| " | " | ||||
| 2.0 | .0315 | .0715 | |||
| 3.0 | .0223 | .0515 | |||
| 4.0 | .0157 | .0357 | |||
| 5.0 | .0111 | .0253 | |||
| 6.0 | .0079 | .0179 | |||
| 7.0 | .0056 | .0126 | |||
| 8.0 | .0039 | .0089 | |||
| 9.0 | .0028 | .0063 | |||
| 10.0 | .0020 | .0045 | |||
| 11.0 | .0014 | .0032 |
Using the value 4 for the solar motion, instead of 3.5, found by Kapteyn, all these parallaxes should be diminished by one-eighth of their amount.
Unfortunately, owing to the great diversity in the absolute brightness of the stars, and the resulting great difference in the distances of stars having the same magnitude, these numbers can give us only
