Rambaut — Detenniiiiny Didcuice of a Double Star. 665 



b' = — — ; we have, therefore, v = ^ And if 



v^ 1 - £- cos'^c/) Fr -v/ 1 - e- cos-^ 



a is the semi-major-axis, measured as usual in seconds of arc, and if 11 

 denotes the parallax, and H the radius of the Earth's orbit, «i= — ; 

 whence we have (r being also expressed in seconds of arc) 



2OTa-y/l - e- . ^ . sin (^ -X) siny 

 Fr ^1 - e- cos^^ 



In this formula the unit of time is a year. So that if V is ex- 

 pressed in miles per second, we must divide the right-hand side by 



2t^R - . 



365 X 24 X 60 X 60 = s. Then = 1 is the mean distance traversed 



s 



by the Earth per second, and we have finally 



la^\/l -e^ sin (<^ - A) sin y 

 Fr\/l-e^ cos-^ 



h-x' 

 To find the angle <f>, we have tan ^ = — ^— ,, cc' and y' being the co- 



ci y 



ordinates of the position in the orbit referred to the centre and axes of 



the ellipse, or tan<f> = — . , x and y being the co-ordinates of the 



same poiat referred to the major axis and the latus rectum. Therefore 



tan(^ = -(l-e2) ^—K-' 



The equations necessary to find the parallax are, therefore, the 

 ordinary equations of elliptic motion — 



( 1 ) u - € simi = nt, 



(2) *^°2 = >/r^^'^i' 



(3) r - a(l -e cos w) ; 



in which a, e, n, t denote respectively the semi-axis-major, the eccen- 

 tricity, the mean angular motion, and the time from periastron ; while 

 r. 0, and u denote the radius vector, the true anomaly, and the eccen- 

 tric anomaly ; and 



, , „, r cos 6 + a€ 



(4) ta.^ = -(l-r)— ^j^; 



