68 Central Forces. 



tide 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. t\/A, 



(13); hence -77=V=the velocity, (at the distance ?",) =flV A X 



sin./;^^A, and a— r— a versin. ^ v^A = the space fallen through; 



iV^A being an arc, such that cos.2;VA=-' or ^v A = arcf cos.=-|. 



(Prin. B. I, prop. 38.) 



Again, it is evident by (13), that vi^hen r=0, cos.^v A=0; 



— P P 



.'. t's/A= ^ or t= ,- , (14); the time t as given by (14) is the 



time of the descent of the particle from the distance a to the centre 

 of force ; but as the value of t does not involve a, t will be the same 



T 



whatever « maybe; by (12) and (14), 1 have i=^' (Prin. cor's 1 



and 2, same prop.) It is evident by (13) that when z;v^A=P, or 



P 



t=—T=-, (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 return 



T P 



from the distance — a to the distance -\-a, in the time t='^= ,— , 



and that it will oscillate after the manner of a pendulum, the time of a 



2P . . . 



whole descent and its subsequent ascent being T= —7= : if F is given 



at the distance a, then F'=Aa, or A= — , hence T=:2Px V/ — , 



(16), is a formula by which T is easily found ; F'=the value of F 

 at the distance a. By substituting the value of ;--, as given by (6), 



in (7); then putting tan. ?; = \/^ X tan. t/, there resuhs dt = 



a 



dt^u.v' i ^^^ -, /l+tan.t>^ 



—Tr= , whose mtegral is t== — -T^Xh.l.l — ■ , 



vAx(l-tan.^?;') ^ 2VA \l—ian.v'j 



[v, v', t, commencing when the particle is at the extremity of a;) let 

 li.l.e—l, then tan. t;'=— — — 5 .'.tan. v = 'V — X 7=~ > 



