188 



DETERMINATION OF AN ORBIT FROM 



[BOOK II. 



y&quot; 



respectively. 



A new verification of the whole calculation thus far can be obtained from the 

 mutual relation between the sides and angles of the spherical triangle formed by 

 joining the three points D, D, D&quot;, from which result the equations, true in gen 

 eral, whatever may be the positions of these points, 



sin (AD AD ) _ sin (A D A D ) sin (A D A D) 



sine 



sm 



Finally, if the equator is selected for the fundamental plane instead of the eclip 

 tic, the computation undergoes no change, except that it is necessary to sub 

 stitute for the heliocentric places of the earth A, A, A those points of the equa 

 tor where it is cut by the circles AB, AB 1 , A B&quot; ; consequently, the right ascen 

 sions of these intersections are to be taken instead of /, I , T , and also instead of 

 A D, the distance of the point D from the second intersection, etc. 



138. 



The third step consists in this, that the two extreme geocentric places of the 

 heavenly body, that is, the points B, B&quot;, are to be joined by a great circle, and 

 the intersection of this with the great circle A B is to be determined. Let B* be 

 this intersection, and d its distance from the point A ; let a* be its longitude, 

 and ft* its latitude. We have, consequently, for the reason that B, B*, B&quot; lie in 

 the same great circle, the well-known equation, 



= tan ft sin (&quot; - - a*) tan ft* sin (a&quot; ) + tan ft&quot; sin (a* a), 

 which, by the substitution of tan / sin (a* I ) for tan ft*, takes the following 

 form : 



= cos (a* f) (tan ft sin (a&quot; ?) tan ft&quot; sin (a ? )) 



_ s in (a* t) (tan ft cos (a&quot; f) -f- tan / sin (a&quot; a) tan ft&quot; cos (a 

 Wherefore, since tan (a* f) = cos / tan ((? - - 0) we shall have, 



tan(&amp;lt;T a) = 



tan (3 sin (a&quot; I ) tan ft&quot; sin ( Q 



cos / (tan cos (&quot; / ) tan 0&quot; cos (a J )) -f sin / sin (a&quot; a) 



