.* ae 
parabola coincides with its axis ; ; hence supposing the pai 
cede from the centre of force, it may be considered ag m 
axis of the ellipse at the distance (r) from the centre of - ; 
_ the distance (r) from the centre of force in the axis produced, in the 
case of the hyperbola; and at the distance (r) in the parabola; 5 
dr 
hence V=—) by substituting this value of V in (10), o (12), 
Witt 
and b = aes 
_ and by reducing they become. 5 V'dt (13), V Qari 
| «Wat ( 14), Sov 15). It may be well to observe that Ve 
“ee vi V’, V", &c. are not supposed to have the same values in the three 
_. eases treated of; but supposing them to be adapted to any one case, 
their values are wiped to’be altered when they are applied to to the 
2a— 
other cases, so as to suit those cases a Pat cot. p= feat 
(16), cot. of pan /2048 ae? (17), and cot. ay? 2P (48), then 
rf ‘ ‘d aV” 
(13), (14), (15) are easily changed to " Sone Ps tay xdt 
: 2 ‘ F "Wf " WwW 
(19), Tr = odo aM tz 
a - 
‘ Sr? coagiishally 
(21); or by integration (19), (20), (21) become 3S 
Sg cosec.?9 dni ae Sr? cosec.*9"dp” 
2 2 gd 
r? cosec. 20/do! os 
5 =F Xdt (20), 
=— Xt (23), = 
s sala the sign of integration, eid it may be observed 
tion is necessary supposing t to commence when r=0. 
: r? Sa ec.*odp 
ii 2a=AB,r cosec. 9=BD, and —— = 
2 20d 
_. pies So ev= the area BD, 
Central Forces. 293, 
(24) indicate Newton’s constructions, (Prin. Vol. I. Sec. 
: P nia 
(22) gives his case 1. for (see his fig. 1,) g—9= his 
ont 
a 
