90 PROCEEDINGS OF THE AMERICAN ACADEMY 



For all the stars we shall need to take the sum 



2 -aZ (siu % cos {}p' — \p)) — x-^sin^ ■^. 

 But, 



2 (sin X cos (V*' — V')) 



S sin- X 



■■ •/., or Z (sill ;f cos (xjj' — ip)) =>! 2^ sin';f ; 



hence, for ?^ stars the sum of squares n of proper motions, each taken 

 equal to unity, will be diminished by jj^^sin^/, or about 0.24 n: 

 as ■/.- = 0.38, and the average sin'"/ is about 0.G4. 



The conclusions I would draw from the investigations so far are 

 these : — 



1. In studying the solar motion, the distances must be assumed with 

 reference to the amount of proper motion, and (approximately) in 

 inverse projwrtion to it. 



2. The smaller proper motions (0".13 or less annually) need careful 

 study at this time. 



3. There is some hope of using the solar motion as a sort of base to 

 advance our knowledge of stellar distances. 



4. The parallaxes of all stars whose parallax exceeds 1" annually 

 (about 60 in number) should be systematically determined by a co- 

 operative arrangement. 



