CHAPTERS ON THE STARS. 457 



in the paper already quoted; and by a process too intricate to be de- 

 tailed 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 mo- 

 tion of the stars. His definitive result is: 



Average speed of a star in space 

 — Speed of solar motion x 1.86. 



This I shall call the straight-ahead motion of the star, without re- 

 gard 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 x 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 x 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 mo- 

 tion 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 direc- 

 tion 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 alge- 

 braic 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, r, 

 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 



