198 DETERMINATION OF AN ORBIT FROM [BOOK II. 



tical, and we should have x = co , \p 1 = 0, and q, therefore, indeterminate. 

 In this case, t,&quot; and r&quot; will be determined, in the manner we have shown, but 

 then and r must be obtained by the combination of equation VII. with VI. or 

 IX. We dispense with transcribing here the formulas themselves, to be found 

 in article 78; we observe, merely, that in the case where AD d is in fact 

 neither = nor = 180, but is, nevertheless, a very small arc, it is preferable 

 to follow the same method, since the former method does not then admit of the 

 requisite precision. And, in fact, the combination of equation VII. with VI. or IX. 

 will be chosen according as sin (AD&quot; AD ) is greater or less than sin (AD (T). 



In the same manner, in the case in which the point Z&amp;gt; , or the one opposite to 

 it, either coincides with B&quot; or is little removed from it, the determination of &quot; 

 and r&quot; by the preceding method would be either impossible or unsafe. In this 

 case, accordingly, C and r will be determined by that method, but C&quot; and /&quot; by 

 the combination of equation VII. either with V. or with VIII., according as sin 

 (A&quot;D A&quot;D r ) is greater or less than sin (A D 1 d&quot;}. 



There is no reason to fear that D will coincide at the same time with the points 

 J5, B&quot;, or with the opposite points, or be very near them ; for the case in which 

 B coincides with B&quot;, or is but little remote from it, we excluded above, in article 

 138, from our discussion. 



144. 



The arcs and C&quot; being found, the positions of the points C, C&quot;, will be given, 

 and it will be possible to determine the distance CO&quot; 2/ from , &quot; and t . 

 Let u, u&quot;, be the inclinations of the great circles AB, A&quot;JB&quot; to the great circle CO&quot; 

 (which in figure 4 will be the angles C&quot;CD r and 180 -- CC&quot;D, respectively), 

 and we shall have the following equations, entirely analogous to the equations 

 3-6, article 137 : - 



sin/ sin (&quot; + ) = sin \ e sin * (f + &quot;), 

 sin/ cos i (u&quot; -(- u) = cos t sin (c f&quot;), 

 cos/ sin k (u&quot; u) = sin $ e cos i (C + &amp;lt;&quot;&quot;)&amp;gt; 

 cos/ cos (u&quot; u) = cos J e cos i (t C&quot;). 



