106 SOME METHODS OF COMPUTING THE RATIO OF 



• 



not employing these equations in parabolic orbits for the final value 

 of Q, where accuracy is desirable ; though they are sufficient for de- 



(65) termining ^"^ and ^, which is the object at present in view. 



(66) When the heliocentric motion is very small, and t" and t near- 

 ly equal, J may be neglected in (53), which becomes 



/f:^\ / sin. &' t"2 ^ / 1 1 \ 



Since g' and r' are necessarily positive quantities, r' will be greater 

 or less than R, according as 0' and 6' have the same or contrary 



(68) signs ; that is, as the apparent path of the comet is convex or 

 concave towards the sun, a well-known connection between the 



(6Sa) apparent and true orbits. (67) is another expression for the equa- 

 tion for finding g' in the method of Laplace ; and under different 

 forms it is used in all the differential methods. 



By the conditions (GG), without neglecting J, using (8), (53) 



(69) becomes p' = rt-[-A) a and b being known quantities. And tak- 

 ing 8' for the angle between g' and R, and z for that between 



(70) p' and r', then o' = ■"'"": '" + ^' and r' = :^i^', which give 



\'^/ S ' 5 ain. 2 sin. 2' ° 



Ji'sin. (»'+») _| 1 sin, 3 z . 



(71) ein. J — '' r "iJi 3 sin.3 S' ' 



from which the unknown quantity z is found by trial, and thence 

 g, g, g", &c. 



The quantities P and Q, employed by Gauss, B. [5999] (38), 

 (39), (235), and (256), may also be used with (52), giving 



In which the values of P and Q are approximately Q—tt" and 

 P = -!^. 



T 



(73) The equations (67) and (71) are approximations only w^hen 

 the heliocentric motion is very small ; in other cases it will be 

 necessary to proceed as indicated in (61). 



In order to correct the assumed values of ^^, ^SlU], in the par- 



