400 ASTROMETRY 



are directed at random, that is, they show no preference for any 

 particular direction." I shall further on refer to this hypothesis as 

 the fundamental hypothesis. 



I shall presently enter into a discussion of it. 

 If, for the moment, we adopt it, we see at once that we can get the 

 mean parallactic motion of any large group of stars free from any 

 admixture of the peculiar motion. 



In Fig. 1, P, I have schematically represented the peculiar motion 

 of a number of stars crowded together near the point S of the sphere. 

 They must show no preference for any direction. As a consequence, 

 the sum of the projections on any line, counted positive one way and 

 negative in the opposite direction, must be zero. 



Now the peculiar motion cannot be observed, for, in addition to it, 

 the stars must have a parallactic motion which is no other than the 

 sun's motion reversed. For all the stars at S this motion is directed 

 along the line Sx towards the Antapex. 



For the star whose peculiar motion is SB, let S(3 be the paral- 

 lactic motion. The total proper motion, which is no other than the 

 really observed motion, will be Sb, the resultant of SB and Sfi. In 

 the same way the observed proper motion of the star, whose peculiar 

 motion is SC will be Sc, etc. The observed proper motions corres- 

 ponding with the peculiar motions in Fig. 1, P, have been represented 

 in Fig. 1, Q. 



The mode of their generation from the two components proves that 

 the sum of the motions projected on Sy at right angles to Sx will be 

 the same as in Fig. 1, P. It must be zero. In the degree to which 

 this condition is satisfied in different parts of the sky there is a pre- 

 cious partial test of the validity of our fundamental hypothesis. 

 We shall revert to it. 



On the other hand, the sum of the observed motions projected on 

 Sx will be the sum of the total parallactic and the projected peculiar 

 motions. The latter being zero, for the same reason as before, we 

 see that, we get the sum of the parallactic motions, consequently 

 the mean parallactic motion of the group free from the peculiar 

 motions. This mean parallactic motion at once yields the mean 

 parallax of the group. 



Adopting Campbell's velocity of the solar system, we have but to 

 divide by 4.20 multiplied by the sine of the angular distance of the 

 group from the Apex. 



In applying this method, however, we shall always have to bear 

 in mind that it rests on the supposition embodied in our funda- 

 mental hypothesis, and that it cannot be used therefore for groups 

 in which the proper motions must evidently favor some particular 

 direction. 



So we can safely apply it to stars of any one particular magnitude. 



