190 DETERMINATION OF AN ORBIT FROM [BOOK II. 



circle BB&quot; itself would become indeterminate. Wherefore we exclude this case, 

 also, just as, for reasons similar to the preceding, those observations will be 

 avoided in which the first and last geocentric places fall in points of the sphere 

 near to each other. 



139. 



Let C, C , C&quot;, be three heliocentric places of the heavenly body in the celestial 

 sphere, which will be (article 64, III.) in the great circles AB, AB, A B&quot;, respec 

 tively, and, indeed, between A and B, A and B , A&quot; and B&quot; ; moreover, the points 

 C, C , C&quot; will lie in the same great circle, that is, in the circle which the plane 

 of the orbit projects on the celestial sphere. 



We will denote by r, r, r&quot;, three distances of the heavenly body from the sun ; 

 by Q, (/, (/, its distances from the earth ; by R, R, R&quot;, the distances of the earth 

 from the sun. Moreover, we put the arcs C C&quot;, CO&quot;, 00 equal to 2/, 2/ , 2/&quot;, 

 respectively, and 



rr&quot; sin 2/= n, rr&quot; sin 2/ = n , rr sin 2f&quot; = n&quot;. 

 Consequently we have 



/ =/ -f /&quot;, A O-\- CB = d, A + O B 1 = d f , A&quot; C&quot; -f C&quot;B&quot; = d&quot; ; 



also, 



sin 8 sin A C sin OB 



~r~ Q Ji 



sin 8 _ sin A C _ sin O B 



sin y _ sin A&quot; C&quot; _ sin C&quot;B&quot; 



i Q&quot; R 



Hence it is evident, that, as soon as the positions of the points O, C , C&quot; are known, 

 the quantities r, r, r&quot;, Q, Q , Q&quot; can be determined. We shall now show how the 

 former may be derived from the quantities 



from which, as we have before said, our method started. 



