SECT. 2.] TO POSITION IN SPACE. 87 



its distance from the heavenly body. Then, ^V denoting an arbitrary angle, the 

 following equations are obtained without any difficulty : 



R cos (L N) + d cos ft cos (I N) = cos B cos (LN) + n cos b cos (lN) 

 R sin (L N) -f S cos sin (X N) = li cos .Z? sin (LN) + cos 5 sin (lN) 



d sin = 7? sin Z* -|- TT sin i. 

 Putting, therefore, 



L (RsmB-\- it sin b) cotan =ju., 

 we shall have 



II. E cos(L N) = RcosScos(L N) + n cosbcos(l N) jUC os(X 

 ILL B sin (I/ N) = cosJB sin (Z N) -\-ncosb sin (lN) p sin (X 

 IV. tf = --,, 



COS (3 



From equations IT. and HI., can be determined R and L , from IV., the inter 

 val of time to be added to the time of observation, which in seconds will be 

 = 493 8. 



These equations are exact and general, and will be applicable therefore when, 

 the plane of the equator being substituted for the plane of the ecliptic, Z, L , I, X, 

 denote right ascensions, and B, b, ft declinations. But in the case which we are 

 specially treating, that is, when the fictitious place must be situated in the eclip 

 tic, the smallness of the quantities B, n, L L, still allows some abbreviation of 

 the preceding formulas. The mean solar parallax may be taken for n ; B, for 

 sin B ; 1, for cos B, and also for cos (I! L) ; L L, for sin (L r --L). In this 

 way, making N=. Z, the preceding formulas assume the following form : 

 I. fi= (RB -(- n sin b) cotan /3 

 II. R = R -f- n cos b cos (I Z) p,cos(l Z) 



T-I-T j-t j- __ n cos b sin (I L) f&amp;lt;sin(l L) 



It? 



Here B, n, L Z are, properly, to be expressed in parts of the radius ; but it is 

 evident, that if those angles are expressed in seconds, the equations L, III. can be 

 retained without alteration, but for II. must be substituted 



TV _ n I n cos b cos (I L) p cos (J, L) 

 &quot;&quot;T 206265&quot; 



