230 DETERMINATION OF AN ORBIT FROM [BOOK II. 



as where the great circle drawn from the middle heliocentric place of the earth to 

 the middle geocentric place of the heavenly body makes a very acute angle with 

 the direction of the geocentric motion, and nearly passes through the first and 

 third places. 



161. 



We will make three subdivisions of the first case. 



L If the point B coincides with A or with the opposite point, 8 will be equal 

 to zero, or to 180 ; y, t , s&quot; and the points I/, D&quot;, will be indeterminate ; on the 

 other hand, /, / , e and the points D, *, will be determinate ; the point will 

 necessarily coincide with A. By a course of reasoning similar to that pursued in 

 article 140, the following equation will be easily obtained : 



, sin (z a) R sin y sin ( A&quot;D 3&quot;) 

 sin z R sin b&quot; sin (AD fl -f a) U 



It will be proper, therefore, to apply in this place all which has been explained in 

 articles 141, 142, if, only, we put a 0, and b is determined by equation 12, 



H /* fi y 



article 140, and the quantities z, r, -, ^, will be computed in the same manner 

 as before. Now as soon as z and the position of the point C have become 

 known, it will be possible to assign the position of the great circle CO , its inter 

 section with the great circle A B&quot;, that is the point C&quot;, and hence the arcs CC , 

 CO&quot;, C C&quot;, or 2/&quot;, 2/ , 2/. Lastly, from these will be had 



_ n r sm2f nVsin 2/&quot; 



: ~~nsin2f T ~ ri smZf 



n. Every thing we have just said can be applied to that case in which B&quot; 

 coincides with A&quot; or with the opposite point, if, only, all that refers to the first 

 place is exchanged with what relates to the third place. 



III. But it is necessary to treat a little differently the case in which B coin 

 cides with A or with the opposite point. There the point C will coincide with 

 A ; /, e, e&quot; and the points D, D&quot;, B*, will be indeterminate : on the other hand, 

 the intersection of the great circle BB&quot; with the ecliptic,f the longitude of which 



t More generally, with the great circle AA&quot; : but for the sake of brevity we are now considering 

 that case only where the ecliptic is taken as the fundamental plane. 



