We now make the general hypothesis that, in the average of a great number of stars, actual motions have no relation to the direction of our sun from the star. Then the components of the actual motion, S D, will in the general average have equal values, positive and negative canceling each other. Hence, if we take the mean of a great number of motions along the apical line it will give us the value of S B due to the motion of the earth, and, hence, the mean parallactic motion of all the stars considered.
The problem now becomes one of averages. We wish to form at least a rude estimate of the average speed of a star in miles or kilometers per second. To show how this may be done let us suppose that we observe the proper motions of a great number of stars at some distance from the solar apex, so that their parallactic motion shall be observable. Stumpe and Ristenpart, the German astronomers, as well as Kapteyn, have considered the relation between the two motions in the following way: We divide the stars observed into classes, taking, say, one class having small, but easily measured, proper motion; another having a proper motion near the average, and a third, of large proper motion. Sometimes a fourth class is added, consisting of stars having exceptionally large proper motions. From each of these classes we can determine, as already shown, the average motion from the direction of the solar apex; that is to say, the average parallactic motion. This will be inversely as the average distance of the stars.
Stumpe's three classes were: I., proper motions ranging from 16" to 32" per century; II., between 32" and 64" per century; III., between 64" and 128" per century; IV., greater than 128". The average of the proper motions in each class, the average of the parallactic motions and the ratio of the two are these:
| Class | Prop. Mot. | Par. Mot. | Quotient |
| " | " | ||
| I. | 0.23 | 0.142 | 1.6 |
| II. | 0.43 | 0.286 | 1.5 |
| III. | 0.85 | 0.583 | 1.4 |
| IV. | 2.39 | 2.057 | 1.1 |
It will be seen that the ratio of the proper motion of the star to the parallactic motion diminishes as the former increases.
The same thing was found by Ristenpart from the proper motions of the Berlin zone, as shown below:
| Class | Prop. Mot. | Par. Mot. | Quotient |
| " | " | ||
| Small | 0.128 | 0.061 | 2.1 |
| Medium | 0.197 | 0.109 | 1.8 |
| Large | 0.374 | 0.279 | 1.3 |
The smaller value of the quotient from stars near to us
