of the Orbits of Double Stars. 43 



we have 



{a + h) sin (s + w) = d sin (/3 + .D)-f c sin (|3 — C) 

 /<:>7\ (a + i) cos (s + w) = <i cos(/3 + D)— c cos(|3 — C) 

 ^^'^ \a-b) sin (s-w) = <^ sin (/3-D)+ c sin (/3 + C) 



(a— 6) cos (s— w) = c?cos(/3— D) — c cos (/3 + C). 



The quantity s derived from these formulae is really not 

 wanted. We may, therefore, likewise make the following 

 transformations : 



(a2_6^) sin 2o> = c« sin SC+d^ sin 2D— 2crf cos 2/3 sin (D+C) 

 i9A\ U'—o) cos2<y = c- cos2C+(f' cos2D— 2co; cos 2/3 cos (D+C) 

 <^^»; ^2^62 =c'+<i*-2ct^cos2(Scos(D— C) 



ab =crfsin2/3sin(D— C) 



in which formulae is contained likew^ise, a test of the accu- 

 racy of the calculation. The ambiguity whether w or 180 + w 

 is to be taken is in the nature of the problem, and is likewise 

 in the formulae (26). It has no influence upon what follows. 

 If we next introduce the quantities d and c into the first and 

 third as also into the fifth and seventh of the equations (25) 

 we obtain, 



RcosP=:X = Mli + |3)+(ccosC COS 2/3-«? COS D) cos (y— «^) 



(ons = -2- (*3+ i^) + (c COS C—d cos D COS 2/3) cos (y + a.) 



^ ' R sin P = Y = i (>n + >i3) + (c sin C cos 2/3-rf sin D) cos (y-«) 



= i («2+«4)+(c sin C—d sin D cos 2/3) cos iy+ct). 



The formulae (28) and (29) do not appear to be very con- 

 venient for logarithmic calculation ; but they might be easily 

 transformed. It has, however, appeared to me that no essential 

 advantages will be obtained from transformations, or ft-om the 

 introduction of new auxiliary angles. The simplicity of the 

 formulae (29) renders it advisable to calculate both expressions 

 for XandY by which all former calculations are checked. 



The projected apparent ellipsis being thus given by R, P, 

 a, b, CO, the real ellipsis, in which the star actually moves, is to 

 be deduced from the two conditions, that the one ellipsis is the 

 projection of the other, and that the place of the star at rest, 

 the point of beginning of our coordinates, is the projection of 

 the focus of the true ellipsis. If we suppose that the plane of 

 the apparent ellipsis passes through the centre of the true 

 one, so that the centres of both coincide, and if we call the 

 radius of the apparent ellipsis which passes through the star 

 at rest /, we shall have this equation : 



(30) -i- = ^ sin (?-«.)"+ -^ cos(P-a))^ 



from which / becomes known. This / being the projection of 

 the scmiaxis major of the true ellipsis, and II the projection 



G2 of 



