156 RELATIONS BETWEEN SEVERAL [BOOK I. 



described about the sun are altogether unknown, it is evident that we cannot dis 

 tinguish between the ascending and descending node. 



It is readily perceived that, just as cos i is the cosine of the inclination of 

 the plane of the orbit to the third plane, so sin ( JV Q ) sin i, cos (N Q ) sin i, 

 are the cosines of the inclinations of the plane of the orbit to the first and second 

 planes respectively ; also that r r sin ( u) expresses the double area of the tri 

 angle contained between the two radii vectores, and zy 1 ys , xz zx , xy yz , 

 the double area of the projections of this triangle upon each of the planes. 



Lastly, it is evident, that any other plane can be the third plane, provided, 

 only, that all the dimensions defined by their relations to the ecliptic, are referred 

 to the third plane, whatever it may be. 



112. 



Let x&quot;, y&quot;, z&quot;, be the coordinates of any third place, and u&quot; its argument of 

 the latitude, r&quot; its radius vector. We will denote the quantities /r&quot;sin(?/ ), 

 rr&quot;sin(n&quot; u},rr sin(u u), which are the double areas of the triangles be 

 tween the second and third radii vectores, the first and third, the first and second, 

 respectively, by , w , ri . Accordingly, we shall have for of , y&quot;, z&quot;, expressions 

 similar to those which we have given in the preceding article for x, y, z, and 

 x f , y , z ; whence, with the assistance of lemma I, article 78, are easily derived the 

 following equations : 



Q = nx n x -\-n&quot;x&quot;, 



= wz tfV + nV. 



Let now the geocentric longitudes of the celestial body corresponding to these 

 three places be a, a , a&quot;; the geocentric latitudes, ft, ft , ft&quot;; the distances from the 

 earth projected on the ecliptic, fT, d , 8&quot;; the corresponding heliocentric longitudes 

 of the earth, L, L , L&quot;; the latitudes, B, B, B , which we do not put equal to 

 0, in order to take account of the parallax, and, if thought proper, to choose 

 any other plane, instead of the ecliptic ; lastly, let D, &, D&quot;, be the distances of 

 the earth from the sun projected upon the ecliptic. If, then, x, y, 0, are expressed 



