240 DETERMINATION OF AN ORBIT FROM FOUft OBSERVATIONS, [BOOK II. 



respectively ^, ^,, whence come 



These auxiliary arcs, which must be taken between and 180, in order that 

 /, r&quot;, may come out positive will, manifestly, be identical with the arcs C B , C&quot;B&quot;, 

 whence it is evident that in this way not only / and / , but also the situation of 

 the points C , C&quot;, are known. 



This determination of the quantities at, x&quot; requires a , a&quot;, b , b&quot;, c, c&quot;, d , tf , Q , 

 Q&quot; to be known, the first four of which quantities are, in fact, had from the data 

 of the problem, but the four following depend on P , P&quot;. Now the quantities 

 P 1 , P&quot;, Q , Q&quot;, cannot yet be exactly determined j but yet, since 



TTT iy _ &amp;lt;* 01) 



-7=7PF)&amp;gt; 



TV P&quot; f ~ f 

 -&quot; 



V. Q -- - } Jck (f--f) (f if) ^ (]? oi) (, 12) cos 1 (v 1 v ) cos (&quot; ) cos 1 (i* v 7 ] 



VI. Q&quot; -- = J kk (f t)(t&quot;t&quot;}7X 



7X ( 7 12) (, 23) cos 1 (J v ) cosi (v&quot; - v ) cos i (&quot; 



the approximate values are immediately at hand, 



&quot; f &amp;lt; 



Q 1 = i Jck (f t) (f if), Q! =tkk(f t ) (f i&quot;), 

 on which the first calculation will be based. 



169. 



The calculation of the preceding article being completed, it will be necessary 

 first to determine the arc C C&quot;. Which may be most conveniently done, if, as 

 in article 137, the intersection D of the great circles A C B , A C&quot;B&quot;, and their 

 mutual inclination shall have been previously determined: after this, will be 

 found from e, C D = z -f- B D, and C&quot;D = z -f B&quot;D, by the same formulas 



