THE DISTANCES OF A COMET FROM THE EARTH. 



109 



which a and a are the geocentric, and C and 

 C" the heliocentric places of the comet, and G, 

 G", those of the sun ; the angle at £, and E a, 

 Ea", are known, and if from (94) are found 

 z^C a and z" = C a", then there are given the 

 two sides EC and EC" and the angle at E, to find 

 the side CC'^v" — v. z and 2" are found from g 

 and p" by trial from the equation ^'"•'^+'' = i., &c., sin. z=.——^ &c. (94) 



* •' * 8in. 2 R ' r 



Otherwise, when right ascensions and declinations are employed, 

 the included chord may be used. 



<?={x"—xy + {y" — yf + (z" — z)\ this form being susceptible (9^) 

 of more accurate computation from the tables, though it is less 

 convenient than the simpler expression which may be derived from 

 it, B. [5994] (106), &c. ; x, y, and z here represent the heliocentric ('JC) 

 coordinates of the comet. The assumed value of q' (76) 



q'{2-q') = 



ir + riif 



sm} n, il-q') = cos.n,q'=i{^y =2 Sin:- in, (0' 

 substituted in Gauss's equations B. [5995], (28), (39), [5997] 

 (101), &c., gives, by combining in one equation the expressions 

 for t' in all the conic sections. 



') 



^2" 



(r^ 



+ 2sin.2i rf 

 ^ — 2 sia.2 J h' sec. 



;.') L^ + Slr::^; _ 



2 sin.2 i g' \ G' 



2 sin.2 * A' sec. A' 



)Zl (^«) 



in which G'^[l-{-j\sm:g'+^sm.* g'-{- &c.], and H'=[l— f^y tan.^ A'+ (oo) 

 /^ tan.* h' — &c.]. The values of G' and H' being given in Ta- 

 bles I. and II. (98) contains but one unknown quantity, g' when 

 the values of r, r", and q' are elliptical, h' when they are hyper- (lOO) 

 bolic, and g' = h' =zO when they are parabolic. The quantity 

 within the brackets in (98) is the coefficient of [r r"] in the equa- 

 tion t' -/^ ^= [rr"] y, where p is the semiparameter and (loi) 



y, _ Pi I 9 ( 1' +2sin.2ig' \a>-y 



J — L ~r TT Vl— 5" — 2sin.2i/('sec. A7 //ij' K^^-') 



which can be found from (98) only by approximation. 



