Central Forces. 341 
distance will approach the centre of force, and after describing an 
angle=— will be at a distance=R from the centre of force, and that 
the portion of the curve described in passing from the least to the 
greatest distance is equal and similar to the portion described in pass- 
ing from the greatest to the least distance, and that these portions are 
similarly situated on opposite sides of the greatest distance ; also 
after having arrived at the distance R, the angle Will be repeated 
Qe’ ‘ 
and the particle will arrive at the greatest distance R+oa when it 
will again approach the centre as before, and so on perpetually. Put 
e’ 
R+-7=R, and mi Ri then (17) becomes r=R’(1 — e cos. mv), 
(19), which is the equation of the curve described by the particle. 
/2 Nill 
Pot =e , Vf =n, then by (19) neglecting quantities of the 
| / 
order e?, r? = R/2(1 —2e cos. mv) =-(1 — 2e cos. mv), hence and by 
(14) there results ndt=dv —2e cos. mvdv whose integral gives nt= 
F 2e sin. mv 
; which needs no correction, since ¢ and »v commence 
at the origin; or mv=mnt+2e sin. mv, and by neglecting quantities 
in. mnt ; 
of the order e2, vento _ , (20), and r=R‘(1 —e cos. mnt), 
(21); (20) and (21) are sufficient to find the place of the particle 
at any time (t). Let v° denote the degrees in v as given by 
eR, agit Siam 
RYR ~ dhlgR 
1g004 / oR 2 190°4/ 2ER | (93); (22) evidentl 
0 V hore =180 VS a 3); (22) y 
agrees with the method given by Newton in the ninth section of the 
rincipia, for determining the angle between the apsides in orbits, 
which differ very little from circles; but it appears to me that (23) 
will generally be more convenient in practice and it may be observ- 
ed that by neglecting quantities of the order eR, in comparison with 
1 
(18), then since a ; (18) becomes v°= 
n 
or Ar 
R, we may write r instead of R in (22), (23). Let gan ps A= 
