SECT. 2.] TO POSITION IN SPACE. 91 



both of which, in the same manner as V., may be rendered more simple by the 

 introduction of auxiliary angles. The solutions resulting from the preceding 

 equations are met with in VON ZACH MonatKche Correspondenz, Vol. V. p. 540, col 

 lected and illustrated by an example, wherefore we dispense with their further 

 development in this place. If, besides u and r, the distance J is also wanted, it 

 can be determined by means of equation in. 



f 



75. 



Another solution of the preceding problem rests upon the truth asserted in arti 

 cle 64, III., that the heliocentric place of the earth, the geocentric place of the 

 heavenly body and its heliocentric place are situated in one and the same great 

 circle of the sphere. In fig. 3 let these places be respectively T, G, H further, 

 let & be the place of the ascending node ; 8, T, 0,11, parts of the ecliptic and 

 orbit ; GP the perpendicular let fall upon the ecliptic from G, which, therefore, 

 willbe=;. Hence, and from the arc PT=L /will be determined the angle T 

 and the arc TG. Then in the spherical triangle Q, HT are given the angle Q = i, 

 the angle T, and the side 8T:=Z Q, whence will be got the two remaining 

 sides &H= u and TH. Finally we have HG = TG TH, and 



_Rs,mTG . RsinTH 



~^m~H&&amp;gt; * ~~ smffG 



76. 



In article 52 we have shown how to express the differentials of the heliocen 

 tric longitude and latitude, and of the curtate distance for changes in the argu 

 ment of the latitude u, the inclination i, and the radius vector r, and subsequently 

 (article 64, IV.) we have deduced from these the variations of the geocentric 

 longitude and latitude, I and I : therefore, by a combination of these formulas, d I 

 and &amp;lt;\l will be had expressed by means of dti, di, d&, dr. But it will be worth 

 while to show, how, in this calculation, the reduction of the heliocentric place 

 to the ecliptic, may be omitted in the same way as in article 65 we have 

 deduced the geocentric place immediately from the heliocentric place in orbit. 

 That the formulas may become more simple, we will neglect the latitude of 



