98 SOME METHODS OF COMPUTING THE RATIO OF 



the following well-known relation, B. [5994] (266) to (279), de- 

 pending on the supposition that r, r, and r" lie in the same plane : 



(2) = [/-' /'] z—[r r"] z' + [r r'] z". 

 Representing by g, 6, and a, R, &, and ©, the polar coordinates 



(2a) of the comet and sun from the place of the first observation, and 

 accenting the same quantities for the second and third places, 

 g, R, &c., being the distances of the comet and sun respectively, 



(3) and substituting c = p sin. 6 — R sin. in (2), it becomes 



(4) = "f^^S sin. d-'-^.Q' sin. 6' + g" sin. 6" - ^^ K sin. G> + t^ J?' sin. 0' 



— R" sin. 0". 



Making [^ = ™ + J, and ^] = [-^ + ^', and supposing R, R, 

 and R' to lie in the same plane, the last three terms of (4) are 

 reduced to two by B. [5994] (362) ; and (4) becomes 



(5) = ^ Q sin. ^— '^ o' sin. & + o" sin. ff' — jR sin. O -\- J' R' sin. 0'. 



If t, i, and f denote the times for which a, a', and a", &c., are 



(6) given, and t = A: (t" — i), x =k (t" — t), t" = k (t' — t), in which 

 log. A: = 8.2355814, B. [5994] (319) to (360), we have, by neg- 

 lecting the powers of t, &c., above the second : 



(7)'il^ = I,[l-^(.^-x-) + &c.],'l^;==i;[l-^;^(.--r-) + &c.] 



8) ^=l.L(x=_x-) (l,-;L)+,&c.,and^' = i:.l(x--.-) i^-^)+ &c. 



T^ T'^ &c., are supposed (1) to be small quantities of the second 

 (9) order ; therefore ^^ = ir, and '■^^ = ^, may be assumed as ap- 

 proximate values. 

 (10) By putting the angles 6, 6', &c., in (4) successively, either sep- 

 arately or in pairs = 0, that is, by changing the position of the 

 plane of z (2), any term of (4) , or any two together, can be made 

 to disappear. But since the expressions thus derived from (4) wall 

 contain angles referred, in each equation, to a separate system of 



