SECT. 1.] THREE COMPLETE OBSERVATIONS. 187 



These definitions being correctly understood, it will be possible conveniently 

 to distinguish, loth intersections of the two great circles from each other. In fact, 

 in one the first circle- tends from the inferior to the superior hemisphere of the 

 second, or, which is the same thing, the second from the superior to the inferior 

 hemisphere of the first ; in the other intersection the opposite takes place. 



It is, indeed, wholly arbitrary in itself which intersections we shall select for 

 our problem ; but, that we may proceed here also according to an invariable rule, 

 we shall always adopt these (D, D 1 , D&quot;, figure 4) where the third circle A&quot;B&quot; passes 

 into the superior hemisphere of the second A I?, the third into that of the first 

 AB, and the second into that of the first, respectively. The places of these inter 

 sections will be determined by their distances from the points A and A&quot;, A and 

 A&quot;, A and A , which we shall simply denote by A D, A&quot; I). AD , A&quot; I) , AD&quot;, AD&quot;. 



Which being premised, the mutual inclinations of the circles will be the angles 

 which are contained, at the points of intersection D, Z&amp;gt; , D&quot;, between those parts 

 of the circles cutting each other that lie in the positive direction ; we shall 

 denote these inclinations, taken always between and 180, by e, F , a&quot;. The de 

 termination of these nine unknown quantities from those that are known, evi 

 dently rests upon the problem discussed by us in article 55. We have, conse 

 quently, the following equations : 



[3] sin * s sin i (A D -f A D) = sin } (f f) sin * (/ -f /), 

 [4] sin e cos * (A D -f A D) = cos (t f f) sin } (/ /), 

 [5] cos J sin * (A D A&quot;D) = sin } (f 1 I) cos } (/ -f /), 

 [6] cos } cos (AD A D) = cos } (f f) cos i (/ /). 



J (A D-^-A&quot;D) and sin E are made known by equations 3 and 4, I (A D A D) 

 and cos i e by the remaining two ; hence A D, A&quot;D and e. The ambiguity in the 

 determination of the arcs (A D -\- A D), (AD A D), by means of the tan 

 gents, is removed by the condition that sin f, cos f, must be positive, and the 

 agreement between sin e, cos t, will serve to verify the whole calculation. 



The determination of the quantities AD 1 , A D , e , AD&quot;, A D&quot;, t&quot; is effected in 

 precisely the same manner, and it will not be worth while to transcribe here the 

 eight equations used in this calculation, since, in fact, they readily appear if we 

 change 



