Solution of a Problem in Fluxions. 331 



I will now notice some things which readily follow from what has 

 been done. Let a, p', be the same as before ; P=3.14159, etc.= 

 semicircumference of a circle rad. (1) ; T= the time of revolution 

 of the particle in an ellipse around a centre of force at the focus ; 

 T'=do, around a centre of force at the centre j the arbitrary constant 



g- being (as heretofore) so assumed as to = the area described by 



(r), the radius vector around each centre of force in the assumed 

 unit of time. At the seventy-third page of the last Journal I found 



F= 3 , =■ the force to the focus of any conic section; suppose 

 FXr2=const- then — =const. ; or p' as c''^ ; (Prin. B. 1. sec. 3. 



prop. 14.) Suppose again that Fxr2=const. then— = const, as 



c'T / a 3 



before; but in the ellipse -^ ='Pv a^p' =the area; hence mj = 



= ^p, = const. ; or T as a" ; (P. B. 1. prop. 15.) By (2) of 



this paper, F=-7-; the force to the centre of an ellipse; suppose 



F c'T' . 4P2a3p/ F 



— =const., then -;^=Pv a^'p', ore' 2= — ^7; — .'.- = const. =; 



4P2 . . . . F 



= rri/g ''- T'=const..*. in different ellipses, if - = const.T' = const. 



the circle is here considered as an ellipse; (P. prop. 10. cor. 2.) 

 Again, if Fxr^ = const, in different conic sections, (as above), 



—7- =const. but —=V the velocity.'. Vis as — ; (P. prop. 16). 



1 2 2c'2 /I 



By (8) of the last Journal, — = — .-.V^ = — 7, or V as v -. (P. 



prop. 16. cor. 6.) By (7) ♦♦ (8) -(9) I have in any conic section 

 -^=-^, but by (3) of last Journal, V : V': : V - : V-^-.-.V: 



V':: V r'p' \p', (P. prop. 16. cor. 9.) Again,p2=-^— r- ; hence, 



V : V: : v'2a+r : Va (10); the sign — being used in the case of 

 the ellipse, and -f- in the hyperbola, and r is to be neglected in the 



