[ 2ZO ] 



now 



I — ecs, a 



= I -\-e cs, a ■^f-e'^ cs', a + et cs\ a + &c. 



•.y= — f '+ e — f ' cs, a + e' cs\ « -f if' a\ a + &c. = 



— — + IZli. cs, a -I tf'fj, 2a-| f' cs, ^a -^ &c. This feries 



24 2 4 - 



.might be continued ad libitum from the relation of the terms to 

 each other, which relation may be found by a method well 

 known, but the above is fufficient for the prefent purpofe. Hence 



y' = — — ■ — cs a ^ — cs 2 a -i <^^ 3^^ &c. 



2 2 2 ' 2 



J, ^ ^ e^ CS, a ^ e^ cs 3 a &c. 



8 4 



&c. &c. 



Prop. 2. The centripetal force varying partly in the inverfc 

 duplicate ratio, and partly in the dired fimple ratio of the diftancc 

 to determine the orbit defcribed, when that orbit has two apfides. 

 (Newtoni Prin. Sed. 9. Prop. 45. Cor, 2.) 



Solution. Let, as before, the great- 

 eft diftance S A = i + f , the leaft dift. 

 S B = I— e, the angle ASP = A, the 

 diftance SP=x, the perp.SY=/, and let the 



force at P be expreffed by — - — ex. Let 



alfo S Z be the height from which a body 

 muft fall to acquire the velocity at A. 



