Central Forces. 67 
ap! 
cor. 1. and scholium.) Put Pp? =r? X 
a? sin.? v + p/? cos.2 0 
Te gl a ae 
: : : a?! 
then (2) is easily changed to r? ~~ (eee (5), and (3) to 
2m/ 
r? et » (6); and it is evident that v= the angle made 
by r and a. 
SS. aides redv 
By substituting in (4), for c’2 its equal di’ have eG Aa®p” 
(7); dt =the element of the time (t), and dv =the element of the 
angle v, supposing v to increase with ¢: I shall also suppose that 
t and » commence when the particle is at the extremity of a. 
Substituting in (7), for r? its value as given by (5), there results 
a? n'dv : p’ 
di. Aich he satttic tangs te ee 
Vv Aa®n’ x (a —(a—p’)cos.?v)’ pier re ve 
vp’ : v 
X tan. v’, (8), becomes dit=——, or by integration t=—>=, (9), 
aN vA 
which needs no correction, for when t=0, v=0, and by (8) when 
=0, v'=0. By (9) v'=¢W A, which substituted in (8) gives tan. v= 
qf aetw A, (10); hence, and by (5), 7? ==@* e038." iw A+ 
ap'sin.? tv A, (11); by knowing when the particle is at the extremi- 
ty of ' its position is easily found at any other time (¢), for v and r 
are easily found at that time by (10) and (11), and hence the place 
of the particle becomes known at the same time. Let P= the semi- 
citcumference of a circle rad.=1, and T'= the time of revolution of 
me particle. Now when.the particle has made a fourth part of a 
. - T r / 
revolution, v=5, and t=, and (10) becomes tan. >= 2 x 
eae om ‘ : 
tan. 7 vA, but tan. a infinity, .*. tan. av A, is also infinite, hence 
p_2P ; : 
mat (12); it is evident by (12) that the time of revolution (‘T) 
will alwa 
(provide 
dent of 
ys be the same, whatever the axes of the ellipse may be, 
d A is invariable ;) because 'T’, as given by (12) is indepen- 
the axes, (Prin. B. 1, prop. 10, cor. 3.) If the semiparam- 
Ser p’, is Supposed to be indefinitely diminished, (a remaining inva~ 
rable,) the ellipse will coincide with its axis very nearly, and the par- 
iss 
