74 RELATIONS PERTAINING SIMPLY [BOOK I. 



there results 



R sin (l L) = A sin (I X) 



/ s in (I L} = J sin (I L) 

 r sin (11)=. R sin (lL). 



The first or the second equation can be conveniently used for the proof of the 

 calculation,, if the method I. or II. of article 62 has been employed. In our 

 example it is as follows : 



log sin (l L} . . . 9.4758653 w / L = 3145 / 26 / .82 

 log 4 9.7546117 



9.7212536 

 log sin (lL) . . . 9.7212536 n 



k 



II. The sun, and the two points in the plane of the ecliptic which are the 

 projections of the place of the heavenly body and the place of the earth form a 

 plane triangle, the sides of which are z/ , R , r, and the opposite angles, either 

 lL, I I, 180 --J + Z, or L I, I I, and 180- -L-\-l; from this the 

 relations given in I. readily follow. 



III. The sun, the true place of the heavenly body in space, and the true place 

 of the earth will form another triangle, of which the sides will be //, R, r : if, 

 therefore, the angles opposite to them respectively be denoted by 



S, T, 18Q ST, 

 we shall have 



sin S s mT s 



/ R 



The plane of this triangle will project a great circle on the celestial sphere, in 

 which will be situated the heliocentric place of the earth, the heliocentric place 

 of the heavenly body, and its geocentric place, and in such a manner that the 

 distance of the second from the first, of the third from the second, of the third 

 from the first, counted in the same direction, will be respectively, S, T, & -|- T. 



IV. The following differential equations are derived from known differential 

 variations of the parts of a plane triangle, or with equal facility from the formu 

 las of article 62: 



