of the Orbits of Double Stars. 103 



and from these equations we derive 



r cos V cos i'= 0{cos w' cos ft cos i' — sin w' sin SI} 



+ rjjsin w' cos SI + cos w' sin ^ cos i'}- 



r sin V cos 2' = — 0{cos co' sin ft + sin w' cos ft cos i'} 



+ rjjcos w' cos ft — sin w'sin ft cos i'}. 



Supposing therefore, 



,^2) 6' sin so' =Z'sin(Q— S3) 



^ b' cos co' cos i' — I' cos (Q— S3 ) 



and introducing the well-known formulee, 



r cos V =: a' [cos E'— sin (p'} 

 r sin V = b' sin E' 



*vhere E' stands for the excentric anomaly in the true ellipsis, 

 we obtain, having regard to (39) 



(44.) 



from which E' may be accurately determined. We then make 



R 



M = E' - sin <p' sin E' = E' - _ sin E' 



(45) i 



T = t-^. 



F 

 where T denotes the time of the perihelion passage. The 

 application to all four observations will furnish a complete 

 check on all the calculations. If it should be deemed con- 

 venient, one may likewise employ the following equations for 

 calculating /, I', Q. 



"2- sm F = — sm (P-co) 



■y cos F = — cos (P— w) 

 , g^ IV sin (Q-P) = - ^a?-b^) sin 2F 



P + l'^ = a' + b% P-P = {a^--b') cos 2F. 

 If, on the contrary, the place is to be determined from the 

 known elements, the constant quantities are to be calculated, 

 with a reference to equations (4.2), according to the method 

 introduced by Gauss: 



sin 71 cos U = cos ft ; sin ?/' sin U' = sin ft ; 



sin u cos U = - sin ft cos i'; sin //cos U'= cos ft cos /'; 



(47) a' 



