OF ARTS AND SCIENCES. 



89 



LUNDAHL'S STARS. 



For values of proper motion between 5".13 and 0".13 annually there 



is no constant deviation of the value ^-^ from 0.G6G, or nearly g ; but 



from 0".13 to 0".09, — that is, the smallest proj^er motions thought safe 

 to use in the original investigations, — its value is about 0.46, with un- 

 usual fluctuations. The last value of each set is by some chance larger 

 than 0.666. 



The proper motion of each star being taken as unity, it will be di- 

 minished in every case by the amount 



cos (xp' — 'ip) — -^j . 



The first term, cos- {ip' — \p), denotes the square of the observed 

 proper motion of each star in the direction of the great circle passing 

 through the star and the. apex of solar motion. This is the dii'ection in 

 which alone the proper motion is affected by the solar motion. The 



tt sin T~i '' 

 cos (xp' — \p) — ~Tif~ is the square of this component 



r 

 after the solar motion —r- has been subtracted, projected upon the line 



passing through the star, in the plane passing through the star, apex, 

 and the eye, perpendicular to the line of sight. The difference of these 

 two squares is the proportionate part of the square of the proper mo- 



tion thus allowed for ; and for each star, making — r— = ■/., a. constant 



is equal to 



2 X sin ;r cos (xp' — xp) — y.^ sin^ ;{. 



