of the Orbits of Double Stars. 45 



The equations (34) and (35) being multiplied together, and 

 (32) being made use of, we have 



(a'3_^y2)2 sin 2ca'^ cos i''^ = {a^-b'f sin 2(ft-co)- 

 and this equation, together with the square of the product of 

 the two equations (33), will give after the substitutions by 

 means of (30) and (31), 



(a--b 'y- sin 2(ft-w)2 = R^ sin 2{F-9,f. 

 Taking the square root of this, and substituting (P— w) — 

 (ft— w) foi* P — ftj we may write the equation thus: 

 { (a°— 62) + Ra cos 2(P— w) } sin 2(ft— <«) = jr R' sin 2(.P—c,) cos 2(ft— «;) 



where the upper and lower signs respectivelj' belong together. 

 From this equation it appears that taking the lower signs 



a^— 6-~R^cos 2(P— «j) is proportional to cos 2(ft, — w), and 



— R-sin2(P— 0)) sin 2(ft— w) 



and equation (37), show that the sum of the squares of these 

 two quantities is equal to Z>'* sin i'*. Hence it follows that 

 b'^ sin P sin 2(ft— w) = — R- sin 2(P— w) 



^^sin e'^cos 2(ft— to) = a^—b^ — R^ cos 2(P — w). 

 The equations may with proper regard to the above equa- 

 tions (28) be thus expressed : 



.^^s b'^ sin i'- sin 2ft = (a^—b') sin 2«;— R^ sin 2P 

 ^ ^ b'- sin i'- cos 2ft = {a-—b-) cos 2w— R sin 2P. 



From these we obtain 2ft, and b'^ sin i'^ or b'^—b'~ cos i'% 

 The equation (36) has already given b'^ + b'- cos i'-; and Z/, i' 

 and ft are thus known. The inclination is obtained by the 

 following rigorously correct equation : 



, -12 _ 2Z>" sin P 



^"g ^ — y-ij^i/i cos i'-' — V^^ sin i'^^ ' 



If we always assume it < 90°, we shall still have the uncer- 

 tainty with regard to ft, whether we must take ft or ft + 1 80°. 

 This uncertainty belongs to the nature of the subject, as 

 the projection leaves it doubtful whether the perihelion, or 

 aphelion is turned towards us. In order to leave, however, 

 no uncertainty with respect to the other angles, it will be best 

 to determine the angle w' by ft, by which it becomes indiflferent 

 which angle has been taken. Calling it the longitude of the 

 perihelion, so that w = £3 + w', we have: 



.„^ ^ a' sin (tt— S ) = —/sin (P— S ) sec i' 



a! cos (tt— £3 ) = —I cos (P— 8 ). 

 We have next, 



(40) sin f' = ^ , cos ?' = ^, , 



by 



