SECT. 1.] THREE COMPLETE OBSERVATIONS. 197 



By combining equations V. and VI. with the following taken from article 139, 

 VIII. / sin ( &quot; A&quot;D -f d&quot;) = R&quot; sin d&quot;, 



IX. r sin ( A D 1 -\- 8) = R sin d, 



the quantities f, &quot;, r, r&quot;,will be thence derived by the method of article 78. 

 That this calculation may be more conveniently effected, it will not be unaccept- 

 able to produce here the formulas themselves. Let us put 



n ^7-1 -^ sin 3 



L i7 J sm(AJy &amp;lt;J) = 



n i -ff sind&quot; _ 



L 10 J sm(A&quot;D d&quot;)~ 



C Q S (AD -&amp;lt;i) 

 L 19 J 5 sin* =A&amp;gt; 



[20 ] ^^--^-l&quot; 



sin i 



The computation of these, or rather of their logarithms, yet independent of P 

 and Q, is to be regarded as the fifth and last step in the, as it were, preliminary 

 operations, and is conveniently performed at the same time with the computation 

 of a, b, themselves, or with the fourth step, where a becomes equal to 4, 

 Making, then, 



nr sine . , , ., ./, 



.- , sin (z -\- A D o ) = , 



i M\ &quot; 



8}=p , 



n sin 



n r sin / 

 n&quot; sin 



we derive L and r from r sin =p, r cos C = q ; also, t&quot; and r&quot; from r&quot; sin &quot;C&quot; =p&quot;, 

 and r&quot; cos &quot; = q&quot;. No ambiguity can occur in determining C and i&quot;&quot;, because r 

 and ;&quot; must, necessarily, be positive quantities. The complete computation can, 

 if desired, be verified by equation VII. 



There are two cases, nevertheless, where another course must be pursued. 

 That is, when the point ff coincides with B, or is opposite to it in the sphere, 

 or when AD $ = or 180, equations VI. and IX. must necessarily be iden- 



