parallactic motion of a star lying at right angles to the direction of the solar motion during the time that the sun, by its proper motion, is passing over a space equal to the radius of the earth's orbit, is equal to the parallax of the star. For this parallax is simply the angle subtended by that radius as seen from the star; and the same angle is the difference in direction of the star as seen from the two ends of the radius.
As yet, the actual amount of the sun's motion has not been well determined. Kapteyn's estimate is 16.7km. per second, which may be called 10 miles. But the results of additional determinations of radial motions make it likely that this result should be increased to perhaps 19 or 20km. per second, or 4 radii of the earth's orbit per year. Accepting this speed we shall have the following rule:
The parallax of a star lying in a direction nearly at right angles to that of the solar motion is equal to one-fourth of its parallactic motion in a year.
In the case of stars in other directions, the parallax would be greater in proportion to the cosecant of the angle between the direction of the star and the solar apex.
If the stars were at rest this rule would enable us immediately to determine the distance of any star by its proper motion, which would then be simply the parallactic motion itself. Unfortunately, in the case of any one star considered individually, there is no way of deciding how much of its motion is proper to itself and how much is the parallactic motion. But when we consider the great mass of stars, it is possible in a rough way to make a distinction between the two motions in a general average.
The direction or motion of any particular star having no reference to that of the sun is as likely to be in the direction of one of the three components we have described as of any other. Hence, in the average of a great number of stars we may conclude that these components are equal.
One of the simplest applications of this law will enable us to compute the mean parallax of the stars whose radial motions have been determined. As this application is, in the present connection, made only for the purpose of illustration, I shall confine myself to the 47 stars of which the radial motions have been measured by Vogel. The mean annual proper motions of these stars, taken without any regard to their signs, are:
| Including Arcturus. | Omitting Arcturus. | |||
| " | " | |||
| In right ascension | 0.163 | 0.144 | ||
| In declination | 0.155 | 0.168 |
The difference of the mean motions in right ascension and
