SECT. 1.] THREE COMPLETE OBSERVATIONS. 191 



140. 



We first remark, that if JV were any point whatever of the great circle CO C&quot;, 

 and the distances of the points C, C , C&quot; from the point N were counted in the 

 direction from to C&quot;, so that in general 



NC&quot; NC = 2/, NO&quot; N0= 2/ , NO N0= 2/&quot;, 

 we shall have 



I. = sin 2/sin NO sin 2/ sin NO -f sin 2f&quot; sin NO&quot;. 



We will now suppose N to be taken in the intersection of the great circles 

 BB*B , CO C&quot;, as in the ascending node of the former on the latter. Let us 

 denote by , Of, &quot;, &amp;gt;, 2) , 2&amp;gt;&quot;, respectively, the distances of the points C, C , C&quot;, 

 D, D , D&quot; from the great circle B B*B&quot;, taken positively on one side, and nega 

 tively on the other. Then sin d, sin , sin G&quot;, will evidently be proportional to 

 smNC, mi NO , sin NO&quot;, whence equation I. is expressed in the following form: 



sin 2/ sin &amp;lt; sin 2/ sin & -f- sin 2/&quot; sin g&quot; ; 

 or multiplying by rr r&quot;, 



II. = nr sin nY sin &amp;lt; + it i&quot; sin &quot;. 



It is evident, moreover, that sin G is to sin 3) , as the sine of the distance of the 

 point C from B is to that of D 1 from B, both distances being measured in the 

 same direction. We have, therefore, 



. ~ sin f sin CB 



- Sin li = 



. , . n , - vr , 



sm (4 D oy 

 in precisely the same way, are obtained, 



sin X&quot;sin OB 



__ 

 &quot; Sin Vi - 



% ~. *~TV/ rr j 

 sin (A D d) 



ff, sinXsinO&quot;7? sin X&quot; sin C B* 



Sin G = 7- 



(sin ^ ZT S + ff) sin (A! If 



_ si 



~ 



sin T sin C&quot;B&quot; 



Dividing, therefore, equation II. byr&quot;sinG&quot;, there results, 



_ rsmOB sm(A&quot;D f ^ ) , SsmC B* sm(A&quot;D # ) , 



