Central Forces. 339 
i A 
fails from rest »=O at the origin, and by (12) c=— >, hence 
VA aD an ce R2 —r2 
{12), (13) become V=R XV RP-7?, 1eRy Ve 
ifthe particle is at rest at the origin, but the force repulsive, the sign 
A Te cere 
of A must be changed; hence cC=Ri» V = Re XV HR, 
r2—R2 
A 
t=Rx 
I will now suppose that or=any function of r, and F=5 —, to de- 
termine the motion ; ; supposing the particle commences its motion at 
night angles to the initial direction of r, and that the curve which it 
describes differs very little from a circle, whose centre is at the’ cen- 
tre of force. Let R=the value of r at the origin, and r=R+a, 
doR 
« being always very small; put a OR, then by Taylor’s theo- 
fem, or by (n) Vol. XX, p. 71, or=o(R+x)=eR+<9’R, neglect- 
tag i of the orders x7, 2°, etc. Ialso have (as heretofore,) 
a (14), t=the time, v=the angle described by the radius 
rector (r) around the centre of force in the time (t), 5 C2 area 
described by the radius vector in a unit of time; I shall suppose that 
‘and » commence when the particle is at the extremity of R, or at 
the origin. Let V/=the velocity of a particle of matter, describing 
®circele around the centre of force at the distance R, V=the ve- 
locity in the curve at the origin, (or at the aoe: R;) then (at the 
origin,) Fais=~, or (R=R°V? ; <-=V", or 2 =R°*V?; 
ct Vs v Ry/R 
"a= Va» put R(1—ys )= =e’ and oR =m? ; then e’ is a very 
at quantity because the described curve differs very little from a 
mele (rad.—R. ) Let {=the angle at which r cuts the described 
Curve, then ae te +) I have by (HH), Vol. XVI. p- 286, 
Poe Seal ? cot. cnibge g oo (5a): or by neglecting the term 
2dr 
