68 es Central Forces. 
ticle may be considered as falling from the extremity of a, in the right 
line a, towards the centre of force, and (11) becomes r=acos. tv A, 
. d : is 
(13); hence c= =the velocity, (at the distance ry) =av AX 
sin.t’ A, and a—r=aversin. t~ A=the space fallen through; 
= oor oss r 
tv A being an arc, such that cos.tV A=" or ¢V A=are (cos.=7). 
(Prin. B. I, prop. 38.) 
Again, it is evident by (13), that when r=0, cos.tV A=0; 
P 
av A’ ) | 
time of the descent of the particle from the distance a to the centre 
of force ; but as the value of ¢ does not involve a, ¢ will be the same 
“tVA= gor t= (14); the time ¢ as given by (14) is the 
whatever a may be; by (12) and (14), Ihave ¢=7- (Prin. cor’s 1 
and 2, same prop.) It is evident by (13) that when «V7 A=P, or 
t= Fi? (15), r=—a; which shows that the particle has descended 
below the centre to a distance, which is equal to the distance a, from 
which it fell above the centre, and the time as given by (15) is twice 
the time as given by (14); it is evident that the particle will returm 
from the distance —a to the distance +-a, in the time i= 5 =e 
and that it will oscillate after the manner of a pendulum, the time of @ 
whole descent and its subsequent ascent being T= a if F is given 
kv @ 
at the distance a, then F’= Aa, or A=, hence T=2P x =? 
(16), is a formula by which T is easily found; F/= the value of F 
at the distance a. By substituting the value of r?, as given by (6), 
. ° n ; 
in (7); then putting tan pasar - x tan. v’, there results dt= 
d tan. v’ 1 v’ 
ar 1 whose integral is t=——— xA_l. iA 
VAX(1—tan. *v’)’ e ye eA ee ’ 
(v, v’, t, commencing when the particle is at the extremity of a5) let 
2t4/A a Be 
h.lex¥; then tan, oes aa. ; = . tan. t= af 2 $4 ¢ see LN 
e +1 ° etV Art 
