in the paper already quoted; and by a process too intricate to be detailed in the present work he has reached certain conclusions as to the ratio of the actual motion of the sun in space to the average motion of the stars. His definitive result is:
Average speed of a star in space
= Speed of solar motion 1.86.
This I shall call the straight-ahead motion of the star, without regard to its direction. But the actual motion as we see it is the straight-ahead motion, projected on the celestial sphere. The two will be equal only in cases where there is no radial motion to or from the earth. In all other cases the motion which we observe will be less than the straight-ahead motion. By the process of averaging, Kapteyn finds:
Linear projected speed of a star
= Speed of solar motion 1.46.
This projected motion, again, may be resolved into two components at right angles to each other. It follows that the average value of either component will be less than that of the projected motion. The components may be the motions in right ascension or declination, or the apical motion and the motion at right angles to it. In any case, the mean value of a component will be:
Speed of solar motion 0.93.
I have used Kapteyn's numbers to obtain the same relation by a somewhat different and purely statistical method.
Imagine the proper motion of a star situated nearly at right angles to the direction of the solar motion. Although we cannot determine how much of its apical motion is actual and how much is parallactic, we can determine whether its motion, if toward the solar apex, exceeds that of the sun. In fact, all stars the apical component of whose motion is in the same direction and greater than that of the sun, whatever the distance of the star, appear to us as moving toward the apex, a direction to which we assign a negative algebraic sign. All stars moving more slowly than this, or in the opposite direction from the sun, will have apparent motions away from the apex, which we regard as algebraic positive. We can, therefore, by a simple count separate the stars moving in the same direction as the sun, and with greater speed, from all the others.
I have classified the stars in this way not only as a whole, but also with reference to their cross-motion—motion at right angles to that of the sun. That is to say, I have taken the stars whose cross-motion, τ, is 2" per century or less and counted their apical motions as positive, negative and zero. Then, I have done the same thing with cross-motions of 3" or 4", then with cross-motions ranging from 5" to 7", and
