127 



ing the fame perihelion didance. The anomalies are called correfponding, 

 when the times from perihelion in each orbit are the fame. 



Solution. Let « = 4 the true anomaly in the ellipfe to radius unity 

 A = 4- the correfponding anomaly in the parabola, the perihelion dif- 

 tance = i and the femi axis major of the ellipfe = « Then the fluxion 



of the elliptic area from perihelion = a x dift. \ = a x — 



«-\-a lCS,2a ' 



a X 2 — n 



i-\-a,2a 

 I 



^ I -■. 2-n 

 • I \^putting n = — ) = a x ' 



l+"'2« - " 2„,a 



n \ . 2 — n ] I 2 « I 3 



y_ i_„+ \=a X X : X — X + — X 



2cs,'-a\ 2 — 2n' cSf'^a 2 i-« cs,'^a 4 



n 



I A 



X &c. Alfo, the fluxion of the parabolic area = 



I — n ' cs^a "» "■ 



Now the ratio of the area in the ellipfe, to the coteraporaneous area i a 

 the parabola, is the fubduplicate ratio of the parameters, or of S i—n : 



V2~* 



Hence, 7^ x i— T^ ' x /• — = / — ;; — X-/- — + 



I cs,^A . cs,*a i-n 2 cs/a 



«~1* 3 ^ 

 X -/. &c. 



I — n 4 cs,a 



Let the fucceflive fluxions of this equation be taken /fr Salium, making n 

 flow uniformly, « = A and n == 0: 



* Newtoni Prin. Prop. 14. Lib. i. 



