336 Central Forces. 
- finite, (supposing » to commence when 7= infinity,) 1 have ro’= 
/ 
c 
V¢ 
, v= the value of v between 7, and r when infinite ; but rtan.p= 
c! ee a 
1 hence in this spiral o’=tan.-): it Is also evident that if a per- 
€ 
. e . . 
pendicular spe —rtan.-) is erected at the centre of the spiral tor 
ave 
infinite, and a line is drawn through the extremity of the perpendicu- 
lar parallel to r infinite, the line thus drawn is an asymptote to the 
spiral. Ife’? is 2 A, but c’” cosec.?) > A, cis evidently positive; then 
A-¢? « 
-" =R*, (5), (6) become (supposing 7 to decrease) 
by putting 
e —dr 1 —rdr 
ees oe, dt = xX whose integrals are 
Me rr? +R? Je Vr? +R? eae 
c PRVRe+P+R/R\  VR?+R?- VRE 
o= = Khe | ee ge 
Ric r/R?2+R?2+7R v 
(v,¢ commencing when r=R’,) or r= a / / RE FRO) Re. 
This value of r, when substituted in the value of v, gives v in terms 
of ¢ and known quantities, whence r and v are easily found at any 
VR?+R?2—-R 
Vo 
c 
=the time from the extrem- 
time which is less than 
ity of R’ to the centre of the spiral. By taking the integral of 
AERA eR b d . G e oti e x hil. 
Ve raf? +R? etween 7, and7r m nite, =RVo 
ve 
7 
, v= the value of » between r andr infinite ; hence 
by substituting the value of r, in terms of ¢, in v’; the position of r 
infinite becomes known at any time; this curve evidently has an @ 
c’ : 
symptote parallel to r infinite, at a distance RE the perpendicular 
c 
from the centre of the spiral to the asymptote; as is evident by 
making r infinite in (1), which gives dears If 2 is ZA; and 
c 
ce’? cosec.2) Z A, c in (3), (1) becomes negative, and its sig8 ye 
be changed, also the signs of the terms involving ¢ in (5)s (6) a 
pay they become do=777 if 
be changed; hence by putting 
