declension is to be regarded as accidental. The velocity of Arcturus is so exceptionally great that we ought, perhaps, to leave it out in taking the mean.
Now, the mean of the radial motions as found by Vogel is 16 kilometers per second. By hypothesis the actual motion in the radial line is in the general average the same as in the other two directions. We have, therefore, to acquire what must be the parallax of a star in order that, moving with a velocity of 16 kilometers per second, its angular proper motion may have one of the above values. This result is by a simple computation found to be:
| " | " | ||
| From the mean motion in R.A. | 0.049 | or | 0.043 |
| From the mean motion in Dec. | 0.064 | or | 0.035 |
The difference of these results shows the amount of uncertainty of the method. Our general conclusion, therefore, is that the mean parallax of the Vogel stars, which may be regarded as corresponding approximately to the mean parallax of all the stars of the second magnitude, is about 0".04.
We have spoken of the two components of the apparent motion as those in right ascension and declination, respectively. But there is no particular significance in the direction of these coordinates, which have no relation to the heavens at large. For some purposes it will be better to take as the two directions in which the motions are to be resolved that of the parallactic motion and that of right angles to it. That is to say, taking the solar apex as a pole, we conceive a line drawn from it to the star, and resolve the apparent motion upon the celestial sphere into two components, the one in the direction of this line, the other at right angles to it. The former, which we may call the apical motion, is affected by the parallactic motion; the latter, which we call the cross-motion, is not, and therefore shows the true component of the motion of the star itself in the direction indicated.
Kapteyn has gone through the labor of resolving all the proper motions of the Bradley stars given by Auwers, in this way. His assumed position of the solar apex was:
| Right ascension | 276° = 18h. 24m. |
| Declination[1] | +34° |
The radically new treatment found in this paper embraces three points. The first consists in the distinction between the spectral types
- ↑ This work of Kapteyn is yet unpublished. The author is indebted to his courtesy for the manuscript copy, with permission to use it. Kapteyn's researches based on this material are contained in a paper on the 'Distribution of the Stars in Space,' communicated to the Amsterdam Academy of Science, January 28, 1893. An abstract in English is found in 'Knowledge' for June 1, 1893.
