SECT. 2.] OF WHICH TWO ONLY ARE COMPLETE. 237 



B, B , B&quot;, B &quot;, the heliocentric latitudes of the earth, 

 R, R , R&quot;, R &quot;, the distances of the earth from the sun, 



(wOl), (n 12), (n 23), (H 02), (H 13), the duplicate areas of the triangles which 

 are contained between the sun and the first and second places of the heavenly 

 body, the second and third, the third and fourth, the first and third, the second 

 and fourth respectively; (rj 01), (vj 12), (17 23) the quotients arising from the 

 division of the areas i (n 01), i (n 12), i (n 23), by the areas of the correspond 

 ing sectors ; 



,_0 L 12) ,,_(n!2) 

 ~ ~(n23) 



v, v , v&quot;, v &quot;, the longitudes of the heavenly body in orbit reckoned from an arbi 

 trary point. Lastly, for the second and third observations, we will denote the 

 heliocentric places of the earth in the celestial sphere by A , A&quot;, the geocentric 

 places of the heavenly body by B , B&quot;, and its heliocentric places by C , C&quot;. 



These things being understood, the first step will consist, exactly as in the 

 problem of the preceding section (article 136), in the determination of the posi 

 tions of the great circles AC B , A&quot; C&quot;B&quot;, the inclinations of which to the eclip 

 tic we denote by /, y&quot;: the determination of the arcs A = d , A B&quot;= 3&quot; will be 

 connected at the same time with this calculation. Hence we shall evidently have 



/ = v (eY + 2 9 R cos s + R tf} 



r&quot;= y/ (e V 4- 2 Q&quot;R&quot; cos d&quot; -f R&quot;R&quot;\ 

 or by putting ^ -f R cos 8 of, ()&quot; -J- R&quot; cos d&quot; = x&quot;, R sin d = d, R&quot; sin d&quot; a&quot;, 



r = \l (of of + a a ) 



168. 



By combining equations 1 and 2, article 112, the following equations in sym 

 bols of the present discussion are produced : 



= (n 12) R cos B sin (I a] (n 02) (9 cos ? sin (of a) -f- .R cos.B sin (fa)) 

 -f (n 01) Xv&quot; cos (1&quot; sin (a&quot; a) + R&quot; cos &quot; sin (/&quot; a)), 



