338, | Central Forces. 
as the elliptic sector VCR; see also cor. 6, prop. 44: in this case 
(see his fig. 3.) CV=R, VCp=1, vcp=RY« v, CP=Cp=r. Let 
c 
P= the semi-circumference of a circle (rad.=1), then it is evident 
i we. Pe 
by (10) when a Lo=5° OF oe that r is infinite ; also by (1) 
when 7 is infinite a, hence it is evident that this spiral has am 
c 
¢ 
asymptote parallel to r infinite, at a distance ee indeed by sup- 
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 infinite distance when r becomes infinite, which it will be when 
/ 
c 
=. on at hence it is evident that the whole curve has two a 
2RV ec 
symptotes, which are respectively parallel to r infinite in the descend- 
ing branch, and to r infinite in the ascending branch of the curve; 
/ 
Ce 
the distance of the asymptotes from r infinite being gt in both 
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 
; Pe’ 
ascending branch = =e 
. IRV c 
If c’ is supposed to be indefinitely small (5) does not exist, and 
rd; : “ 
(6) becomes dt= Vane 5 (11). In this case it is evident the par 
ticle may be considered as moving in the right line drawn from the 
centre of force through its initial position: its distance from the centre 
of force at any time (t) being denoted by r. By (1!) Vaihe ven 
; dr Vert? tA 
locity=*+7 =———— (12); also let R=the initial value of 1; 
+/ cr? LALY CR? +4 
then by taking the integral of (11) PO es (13), 
the upper signs being used when the particle moves from the eon 
of force, and the lower signs in the contrary case. If the partic 
