338 Central Forces. 



as the elliptic sector VCR ; see also cor. 6, prop. 44 : in this case 



(see his fig. 3.) CV=R, YCp=-v, VCP = ^^% , CP=C»=r. Let 



c' 



P=the semi-circumference of a circle (rad. = l), then it is evident 

 by (10) when — --—v = -^-> or i;= — that r is infinite; also by (1) 



c 

 c 

 a 



when r is infinite p = — r= ; hence it is evident that this spiral has an, 



symptote parallel to r infinite, at a distance =~~7^ ', indeed by sup- 



V c 



posing the particle to descend from r infinite, it will approach the 



centre of force until it arrives at the extremity of R, when it will 



recede from the centre of force on the opposite side of R, and will 



describe the remaining portion of the curve, which is evidently equal 



and similar to the portion in which it descended and will go off to 



an infi.nite distance when r becomes infinite, which it will be when 



— v = — — y-' hence it is evident that the whole curve has two a- 



symptotes, which are respectively parallel to r infinite in the descend- 

 ing branch, and to r infinite in the ascending branch of the curve; 



c' . 

 the distance of the asymptotes from r infinite being =-7--5 in both 



V c 



branches of the curve ; also the angle made by R and r infinite in 



the descending branch — the angle made by R and r infinite in the 



'Pr' 



ascending branch — - 



^ 2RV( 



c 



If c' is supposed to be indefinitely small (5) does not exist, and 



t.rdr 

 (6) becomes dt=^-y^== (11). In this case it is evident the par- 



tide may be considered as moving in the right line drawn from the 

 centre of fbrce through its initial position : its distance from the centre 

 of force at any time {t) being denoted by r. By (11) V=the ve- 



dr V cr^+A 

 locity=J:-i7= —(12); also let R — the initial value of r, 



j-x/cr^+A+v^cR^+A 

 then by taking the integral of (11) t— (13), 



the upper signs being used when the particle moves from the centre 

 of force, and the lower signs in the contrary case. If the particle 



