72 Solution of a Problem in Fluxions. 



C''=C''' or the areas described in equal times by the radii vectores 



are equal, (which is virtually Newton's supposition,) I have 



F : F'::RP^XSP : SG^ (6) the same as Newton has found. 



FxSG3 . FxSG^ . 



Hence I have F^= ppa y cp or F' is as i? p 2 y op ' if then the law 



of variation of F is known that of F'' is found. If C' is not equal to 



C''2 FxSG^ FxSG^ 



C' I have by (5) F'=^ ^ RP^xSP °^' ^^ ^^ Rpa^ gp (re- 

 jecting the invariable quantities C'^ C : they being the same at every 

 point of the described curve,) the same result as by Newton's suppo- 

 sition, that C'^C" as it evidently ought to be. For it is not the ab- 

 solute value of F' that we seek, but its law of variation at different 

 points of the described curve ; which will evidently be the same 

 whatever may be the time of describing C'^ : but by supposing C'''=C' 

 (as Newton does) we determine the law of variation of F' in a very 



simple and elegant manner. Again die form F= — o'^i „2 • ' aj , ) 



dr 

 applies very readily to the case of the particle describing an ellipse 

 about a centre of force at the focus. For let «, h, respectively de- 



note the greater and lesser semiaxes; — =p' the semlparameter ; 



,\h'^=ap'. Then since r= the distance of the particle to the 

 centre of force, 2a — r equals its distance to the other focus, (Vince's 

 Conic Sections, Ellipse, prop. 1.); alsor, 2a — r, make equal angles 

 with the tangent at the place of the particle (Vince's Conic Sections, 

 prop. 3. cor. 2.) ''.{2a — r)sa\.-\>=p" the perpendicular from the 

 other focus to tlie tangent; but r sin. •v|^=^ that from the centre of 

 force to the tangent; .-'.(2ar- r^) sin. ~ ^:=p-p" t^h"^ (Vince's Con. 



1 2 1 



Sec. prop, 6.) =ap'. Hence I have ^2 gi^. 2^=— /-— / C^) ; ^f "^ 



1 1 2 



(7) I omit — , it becomes -ir~- — rr= ~ (8); which is the case of 

 ^ '' ap' r^ sm. ^-^^ W 



the particle moving in a parabola ; the centre of force being at die 



focus, p' being the parameter of its axis ; if I change the sign of — 



12 1 



in (7) it becomes -r—- — 77= —A : (9) ; this applies to the case 



V / y.2 sm. ^^ rp' ap' ^ ^' ^^ 



of the particle describing an hyperbola; the centre of force being 



