SECT. 1.] THREE COMPLETE OBSERVATIONS. 185 



to the fewest possible details. It will be of equally great importance, if it 

 should not be necessary to proceed in every hypothesis as far as the elements, 

 but if their computation might be reserved for the last hypothesis. In both 

 these respects, our method, which we are now about to explain, seems to leave 

 nothing to be desired. 



136. 



We are, in the first place, to connect by great circles three heliocentric places 

 of the earth in the celestial sphere, A, A , A&quot; (figure 4), with three geocentric 

 places of the heavenly body, B, B , B&quot;, and then to compute the positions of these 

 great circles with respect to the ecliptic (if we adopt the ecliptic as the funda 

 mental plane), and the places of the points B, B , B&quot;, in these circles. 



Let a, a , a&quot; be three geocentric longitudes of the heavenly body, /?. /T, /?&quot;, lat 

 itudes ; /, ^, I&quot;, heliocentric longitudes of the earth, the latitudes of which we put 

 equal to zero, (articles 117, 72). Let, moreover, /, /, y&quot; be the inclinations to the 

 ecliptic of the great circles drawn from A, A , A&quot;, to B, B , B&quot;, respectively ; and, 

 in order to follow a fixed rule in the determination of these inclinations, we shall 

 always measure them from that part of the ecliptic which lies in the direction 

 of the order of the signs from the points A, A , A&quot;, so that their magnitudes will 

 be counted from to 360, or, which amounts to the same thing, from to 180 

 north, and from to 180 south. We denote the arcs AB, AB 1 , A B&quot;, which 

 may always be taken between and 180, by d,d , 8&quot;. Thus we have for the de 

 termination of y and d the formulas, 



[1] tany= . 



sm(( /) 



m-i * tan (a I) 



[21 tano = - . 



&amp;lt;- J cos v 



To which, if desirable for confirming the calculation, can be added the following, 



sin d = -!*-, cos d = cos S cos (a I) . 

 sin 7 



We have, evidently, entirely analogous formulas for determining y f , d f , / , d&quot;. Now, 

 if at the same time /3 = 0. cr 1= or 180, that is, if the heavenly body should 



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