of the Orbits of Double Stars. 407 



tin- f + ?) - tangi(E 4 + E 3 -E 3 -E.) 

 W" tangi(E 4 -E 3 -E 2 +E 1 ) 



,^4^ 

 tang (45 



If we make, consequently, 

 (7) 



i (E4+E3-E2-EJ) = y, we have 

 tang |3 = tang (45 + ? ) tang 

 (8) tangy = tang (45 + ?/) tang a 



tangy = tang (45 + ? 2 ) tang/3 



so that, ?, 5u ? 2 being derived from known quantities (B), if 

 one of the quantities a, /3, y is known, the other two will be 

 found in a very simple manner, and consequently all differ- 

 ences between the excentric anomalies ; as we have 



i (E 4 -E,) = y +ft ft (E 3 -E 2 ) = y - 



The equations (7) may likewise be written in this manner : 



sin(/3 a) = tang sin (/3 -fa) 



(9) sin (y a) = tang ^ sin(y-i-a) 

 sin (y-0) = tang ? 2 sin (y + /3) 



Substituting the quantities a/3y in the expression for (1 2 3 4) 

 in (3), we shall find, 



(10) (1234) = ab sin (y a) sin(y + a) sin 2 /3. 



Between the quantities , ? 15 ? 2 , and a, /3, y, there are different 

 relations, which may serve to check or to change the develop- 

 ments. Thus we have : 



(11) tang (45 + ?). tang (45 + ? 2 ) = tang (45 + ^); or 



(12) sin2?- sin2 1 + sin 2 ? 2 = sin 2 ? sin 2 ? t sin 2 ^. 

 Combining these with the equations 



^2^ = cotang (45 + ?) + tang (45 + ?) 



2 tang 2 = tang (45+ ) cotang (45 + ), we obtain 

 tang 2 cotang (45 + ? 2 ) + tang 2 ? 2 cotang (45 4- ?) = tan g 2 ?i 



tang 2? tang (45 -f &)+ tang 2? 2 tang (45 + ?) = tang 2?, ; 

 or introducing the angles a, ft y, 



g tang 2?, tang 2 g, 



tang y tang /3 tang a 



(13) tang 



