Of KEPLER'S PROBLEM. 243 



Having thus found C and D, we have v — 2 -f iv = 

 z + G X -f D X 2 : that is, 



v zz z -\--^- {7 — 6 cof 2 — cof 2z\ X tan ^ 



-] — L_j ^ 10 — 78cofz — 341 cof 2 2—840^32; — 7cof42|A 2 tan 3 -. 



1 9. Having now refolved the problem that was propofed, it 

 remains to apply it to find the true place of a comet in an ec- 

 centric orbit. For this purpofe, nothing more is wanting, than 

 to be able to determine the angle z in the parabola, at any given 

 inftant of time, reckoning from the paffage over the perihe- 

 lion. We (hall here fuppofe, as a matter already known and 

 demonftrated, the theory that is commonly given of a body 

 defcribing a parabolic trajectory round the fun, placed in the. 

 focus : and we fhall alfo make ufe of the aftronomical tables 

 that have been computed, for finding the true place in that tra- 

 jectory when the time is given. It would indeed be eafy for us 

 to deduce the whole of that theory, and to explain the conftruc- 

 tion of the tables, from the fluxional equation, 



r % 'z — 2p 2 x y (1 +y 2 ) 



obtained above : but this would only be to repeat what is al- 

 ready familiar to aftronomers. 



Suppose, then, a body to defcribe the given parabola, by its 

 gravitation to the fun placed in the focus ; and let us compare 

 the motion of the body in the parabola, with the motion of the 

 comet in the eccentric orbit : If two bodies defcribe different 

 conic fections by the action of a central force, tending to the 

 foci of the curves, and varying inverfely as the fquare of the 

 diflance, it is demonftrated, by the writers on central forces, 

 (Vide Newt. Prin. Math. lib. 1. prop. 14.) that they will defcribe 

 areas, in the fame time, that are in the fubduplicate ratio of the 

 two parameters : Therefore, the area defcribed by the body in 

 the parabola, in any given time, will be to the area defcribed 



by 



