72 Solution of a Problem in Fluxions. 
’=C” or the areas described in equal times by the radii vectores 
are equal, (which is virtually Newton’s supposition,) I have 
F : F’::RP2 xSP: SG* (6) the same as Newton has found. 
Rao FxSG:; F.x SG? 
Hence I have F =RP: xSP % F’ is as RP’ xsp’ if then the law 
of variation of F is known that of F’ is found. If C’ is not equal to 
412 3 3 
C” T have by (5) P=on <ROFESE or F’ is as cline -(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 tobe. 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 
: C” 
simple and elegant manner. Again the form F= — gd ( =a) 
dr 
applies very readily to the case of the particle describing an ellipse 
about a centre of force at the focus. For let a, 6, respectively de- 
2 
note the greater and lesser semiaxes; — =p’ the semiparameter ; 
-1.b7=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.); also r, 2a —7, make equal angles 
with the tangent at the place of the particle (Vince’s Conic Sections, 
prop. 3. cor. 2.) .°.(2a—r) sin. )=p” the perpendicular from the 
other focus to the tangent; but rsin..}=p that from the centre of 
force to the tangent; .*.(2ar—r?) sin. ee ae Mead: Con. 
Sec. prop. 6. : ==ap’.. Hence I have + -=— = ay ae if in 
(7) I omit — = it becomes => i =5 (8); which is the case of 
the particle moving in a parabola; the centre of force being at the 
focus, p’ being the ee of its a if I change the sign of ap! 
mn (7) it becomes =——> POR Pe at = (9); this applies to the case 
of the particle describing an hyperbola; the centre of force being 
