SECT. 2.] TO POSITION IN SPACE. 57 



and use always the same formulas for both, while the common form may fre 

 quently require double precepts. 



50. 



The most simple method of determining the position, with respect to the 

 ecliptic, of any point whatever on the surface of the celestial sphere, is by means 

 of its distance from the ecliptic (latitude], and the distance from the equinox of 

 the point at which the ecliptic is cut by a perpendicular let fall upon it, (longi 

 tude). The latitude, counted both ways from the ecliptic up to 90, is regarded as 

 positive in the northern hemisphere, and as negative in the southern. Let the 

 longitude X, and the latitude /?, correspond to the heliocentric place of a celestial 

 body, that is, to the projection upon the celestial sphere of a right line drawn 

 from the sun to the body ; let, also, u be the distance of the heliocentric place 

 from the ascending node (which is called the argument of the latitude], i be the 

 inclination of the orbit, 8 the longitude of the ascending node; there will exist 

 between i,u, fi,&quot;k. 8 , which quantities will be parts of a right-angled spherical 

 triangle, the following relations, which, it is easily shown, hold good without any 

 restriction : 



I. tan (X Q, ) = cos i tan u 



II. tan /3 = tan sin (X Q) 



III. sin {} = sin i sin u 



IV. cos u = cos ft cos (X a ) 



When the quantities i and u are given, X Q will be determined from them by 

 means of equation I., and afterwards ft by II. or by III., if ft does not approach 

 too near to + 90 ; formula IV. can be used at pleasure for confirming the cal 

 culation. Formulas I. and IV. show, moreover, that X Q, and u always lie in 

 the same quadrant when i is between and 90 ; X & and 360 u, on the 

 other hand, will belong to the same quadrant when i is between 90 and 180, or, 

 according to the common usage, when the motion is retrograde : hence the ambi 

 guity which remains in the determination of X 8 by means of the tangent 

 according to formula I., is readily removed. 



8 



