FOURTH SECTION. 



RELATIONS BETWEEN SEVERAL PLACES IN SPACE. 



110. 



THE relations to be considered in this section are independent of the nature of 

 the orbit, and will rest upon the single assumption, that all points of the orbit lie 

 in the same plane with the sun. But we have thought proper to touch here upon 

 some of the most simple only, and to reserve others more complicated and special 

 for another book. 



The position of the plane of the orbit is fully determined by two places of 

 the heavenly body in space, provided these places do not lie in the same straight 

 line with the sun. Wherefore, since the place of a point in space can be assigned 

 in two ways, especially, two problems present themselves for solution. 



We will, in the first place, suppose the two places to be given by means of 

 heliocentric longitudes and latitudes, to be denoted respectively by X, X , (i, ft : the 

 distances from the sun will not enter into the calculation. Then if the longitude 

 of the ascending node is denoted by 8, the inclination of the orbit to the ecliptic 

 by i, we shall have, 



tan /? = tan i sin (A Q, ), 

 tan /? = tan i sin (X & ). 



The determination of the unknown quantities & , tan i, in this place, is referred 

 to the problem examined in article 78, H We have, therefore, according to the 

 first solution, 



tan i sin (A 8 ) = tan /? , 



,, tanjS tanScos(l 1) 



tan*cos(X 8) = - sin( /_^ v - , 



20 (153) 



