Ball — On Stars tiith a Large Annual Parallax. 219 



the parallel is also affected, by refraction : we have thus the expres- 

 sion 



3438 k tan^2 cos^ sin {2rj - p) 



to denote the correction expressed in minutes of arc, which is to be 

 applied to the observed position angle in order to clear it from the 

 effects of refraction. 



The distance of the two stars is also affected to a certain extent by 

 •aberration. The correction to be applied to the observed distance is 



D sin 1" (/ sin S - A cos {S + a) cos 8), 



where i', h, IT, are given in the Nautical Almanac for the day in 

 question. When this correction has been applied to the observed dis- 

 tance of the two stars, we obtain the distance between the mean 

 places of the two stars for the preceding 1st January. 



The position angle of the two stars is also affected by aberration, 

 and the correction to be applied, expressed in minutes of arc, is 



sin (jS"+ a) tan S. 



On account of the motion of the pole, arising from precession and nu- 

 tation, there is a corresponding change in the direction of the parallel, 

 and therefore a change in the position angle ; the correction 



- — - sin ( 6^ + a) sec S 



will make the position angle what it would have been when referred 

 to the position of the pole on the preceding January 1. The quanti- 

 ties g, G, are those given in the Nautical Almanac for each day. 



As the observations are all reduced to the epoch January 1st, 

 1875, a further correction, 



- 0-3342 {t - 1875) sin a . sec S, 



must be applied to the position angle observed in the year t. 



Corrections must also be sometimes applied on account of the dif- 

 ferences between the proper motions of the two stars. Let 



{t - 1875) Aa and {t - 1875) AS 



be the corrections arising from the proper motions of the principal 

 star relatively to the other star, which must be applied to the right 

 ascension and declination of the principal star to bring the place to 

 the date, January 1st, 1875. Then, the correction to be applied to 

 the distance is 



cos^AS (1875 -t)^ cos S sini?Aa (1875 - t), 



while the correction to be applied to the minutes of the position 

 angle is 



3438 ^\np — (1878 - t) - 3438 cos S cos^? ^ (1875 - t). 



