Of KEPLER'S PROBLEM. 243 



Having thus found C and D, we have v zz z -^ w =z 

 z -j- C>. -\- D \' : that is, 



V — z 4--^ J7 — 6 cof z — cof 2zT K tan ? 

 ' 15 t J 2' 



H — ^ J510 — 78cofz — 34icof2z— 84cof3Z — 7cof4z|x = tan^-. 



19. 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 paflage over the perihe- 

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

 demonftrated, the theory that is commonly given of a body 

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

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

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

 jedory 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 jfluxional equation, 



r''z=:2p' X y {i -{-y-) ■ "^vjg 



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

 ready familiar to aftronomers. 



Suppose, then, a body to defcribe the given parabqla, 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 orbrt : If two bodies defcribe different 

 conic fedlions by the adlion of a central force, tending to the 

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

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

 fF/V/f 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 



