r CHAPTERS ON THE STARS. 455 



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 can- 

 celing 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 kilo- 

 meters 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 dis- 

 tance from the solar apex, so that their parallactic motion shall be ob- 

 servable. Stumpe and Eistenpart, 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: 



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 Eistenpart from the proper motions of 

 the Berlin zone, as shown below: 



Class. Prop. Mot. Par. Mot. Quotient. 



The smaller value of the quotient from stars near to us 



