105 



However, as commentators have laboured much in inveftigating* New- 



ton's correftion, it may not be improper to give the following eafy 



method of deriving terms equivalent to the terms of the Newtonian 



feries, which may, at the fame time, ferve to (hew that the inveftiga- 



tion of the angle at the higher focus mufl be as tedious as the direft 



method of finding the angle at the fun. 



Let the angle fought AHP— w, DCA=c, the mean anomaly=m, pig. 3. 



AC=i and CHrr?. Then from the equable defcription of areas, 



m=c-^es, c 



Alfo by the well known theorem for finding the true from the exccntric 



anomaly. 



s/i+e : v/r^ff : : tang. iSCD : tang. ^SHP 



But tang. 2SCD=tang. 180— DCA -co-tang. iDCA 



2 

 and tang. ^SHP=co-tang. iPHA 



Therefore y/i+e : y/i — e : : co-ta. 4c : co-ta. iiv : : ta. jw : ta. k 

 or log. tang, iwnlog. tang. ic+4log. i+e 



I — e 

 Taking the fluxions of the equations thus obtained, making only e, c 

 and w variable, we have 



• 



c+es, c=o 



• • • 



nv _ c e 



s,w SyC I — e 



• •• 

 From which equations the refpeftive values of w, w, w, &c. arc to 

 be obtained when e=o and c=w=m by taking the fucceffive fluxions of 

 thefc equations per Saltum. , 



I. c+«, m=o f 



. . . I therefore c=—^, 



10= c — esm ^ 



-esm and iu=o 



Vol. IX. ( O ) 2. 



• Simpron's Trafls, pag. 49. Horfley's Newton, vol. z. p. 135. Eraerfon's Com- 

 ment. Le Soeur & Jaci^uier. 



