of the Orbits of Double Stars. 105 



Supposing, therefore, 



cos F = — , — Tg R cos (P— w) = h cos H 



;7^^ sin F = -^, R sin (P-c«) = h sin H 



we have the following equation of condition for the maximum 

 or minimum of g : 



sin2(E' + F) = 2/2sin(E' + F-H) _ ' 



in which form it may be best solved by trials. We determine 

 E'+F, and thence E'. 



This equation will always have two possible roots ; but it 

 may likewise have four, whose reality and limits may, ac- 

 cording to a remark of Professor Dirichlet, be most easily as- 

 certained by differentiation. We then come to the following 

 result : Let us assume 



3 3 



- •v/ {h sin H) = w sin n ; y/{k cos H) = m cos n ; 

 then the equation has for ?re>l, two roots, one where sin 

 (E' + F) is positive, another where it is negative. 



But in case ??z< 1 it has four roots within these limits: 

 sin E' + F = + 1 = = m^ sin rf = m^ sin w = + 1. 

 For by substituting these five values for sin (E' + F) succes- 

 sively into the biquadratic equation which we obtain by elimi- 

 nating cos (E' + F), viz. 

 sin(E'+F)*+ 2//sinHsin(E' + F)^ +(7^2-1) sin (E' + F)^ 

 -2h sin H sin (E' + F)-h' sin W = 0, 

 we get the foUov/ing corresponding values for the sum of all 

 the terms, 



+ k^ cos H^ ; -h^ sin H^ ; +h* sin H^ cos H* ; 



- { 1 -m«} /i^ \/ (sin H^ cos H*) ; h^' cos U\ 

 and as there are alternately positive and negative for m<l, 

 we have a demonstration of the reality of the roots and their 

 limits as above assigned. 



It may lastly be desired to refer the position of the plane 

 in which the motion actually takes place, to some fixed and 

 known planes. If we imagine a plane passing through the 

 star at rest parallel to the equator, the solution of the triangle 

 between the pole of the heavens, the pole of the plane of pro- 

 jection (« and 8, or right ascension and declination of the 

 star), and between the pole which is determined by S3 and i', 

 will give the elements which are required. If we express by 

 8 ' the angle formed by the intersection of the plane of mo- 

 tion with that of projection and the circle of declination of 

 the star counted from the north point through the cast point 



N.S. Vol. 11. No. 62. Feb. 1832. P to 



