of the Orbits of Double Stars. 409 



corresponding to the points 1 and 2 is = a* (E 2 E,), 

 the triangle between the radii and the chord of the circle = 

 % a 2 sin (E a EJ ; consequently the elliptical segment will be 

 = \ a b{ (E 2 EJ sin (Eg EJ }; adding the triangle (012), 

 we have, for the double area inclosed between n 2 and the 

 elliptical arc, this expression ; 



(012) + {(,-,) -sin (,- ,)}**. 

 This area is the projection of the elliptical area, described 

 in the proper orbit in the time (t^ ^). The areal velocity 

 being proportional to the time, the projected area will likewise 

 be proportional to the time. Denoting therefore the double 

 areal velocity in the projected ellipse by k, and applying the 

 same formulae to all other combinations of the four points, we 

 have these equations : 



*i) (0 1 3) = ab {2(y ) sin 2 (y 



Jc (* 4 -/ a )-(0 24) = a b {2 (y + )- sin 2 

 k (f 4 _* 3 ) (034) = afl{2(i8 + a) si 

 which, however, on account of the relations (2), form only 

 three independent equations. The fourth depends on the two 

 first, the fifth on the first and third, the sixth on the second 

 and third, and vice versa. It is likewise to be observed, that 

 these equations suppose that in the interval no entire revolu- 

 tion has taken place, a condition which in the actual applica- 

 tion of them will always be fulfilled. 



In these equations the areas of the triangles, and the times 

 are known ; the quantities a b, a, /3, y, may all be regarded as 

 functions of one only of the three quantities , 6 9 y, and there 

 are consequently three independent equations, arid only two 

 unknown quantities. There must therefore exist an equation 

 of condition which must be fulfilled by the four points, if they 

 are to belong to the same projected ellipse. But instead of 

 deriving this equation beforehand, it appears to be more con- 

 venient to make use of its existence in the course of the cal- 

 culation, in order to change agreeably to it a less accurate 

 datum. 



These equations, on account of their transcendental form, can 

 only be resolved by trials. At first sight, one might be inclined 

 to effect this by eliminating the quantities k and ab, which 

 would lead to an equation between known quantities and three 

 different transcendental functions of the form 2 .r sin 2 x. But 

 this method is not advisable, inasmuch as in that case the 



N. S. Vol. 9. No. 54. June 1831. 3 G quantity 



