86 PROCEEDINGS OF THE AMERICAN ACADEMY 



IX. Assuming for the moment that the apex of solar motion is 

 known, and employing Argelauder's notation ; if now we compute 



cos ;f = — sin Z) sin d -|- cos D cos 8 cos (« — A), 



cos D sin (a — A) 



sm w' = : — -, 



^ sin X 



A g sin \fj=. zl a cos 8, 



A g cos wz= A 8; 



(where a, 8, are the star's right ascension and declination, 



A, D, the like co-ordinates of the apex, 



;r the star's distance from the apex, 



\p' the angle of position at the star of the great circle passing 



through star and npex, 

 A a, A 8, the star's annual proper motion in right ascension 



and declination, 

 A g the same in arc of a great circle, 

 \p the angle of position in which the star appears to move) : 



then will each star give an equation ^ sin 'j^ = r A g cos 



{ip'-ip). 



Here ^ is the annual solar motion, and r the star's distance from 

 the sun. 



X. We shall now proceed as follows : — 



Grouping together stars whose proper motions are nearly equal, 



and making within such a group rAg= 1, we shall find from each 



C 

 group a value of ^ expressed in terms of r A g, or — — . If these values 



of — r— for 'widely different values of A g are nearly equal, — as they 



proved to be in the preliminary investigation, — we may conclude that 

 rAgis nearly constant over a wide range of values oi A g ; or, in other 

 words, that star-distances are on the whole inversely proportional to 

 proper motions. 



As the values of sin ;f vary greatly, it is proj^er to find — - from each 



group by least squares ; and I have done so. 



m, n ^ 1 C S [sin ;t cos ((/'' — V')] 

 The formula used was -r^r- = — ^ „ . ., 



