Central Forces. 339 



A 



falls from rest t;=0 at the origin, and by (12) c=— :j37, hence 



Va . /R^-r^ 



(12), (13) become V=^XvR'-?-% ^ = RxV — j-—; 



if the particle is at rest at the origin, but the force repulsive, the sign 



• A a/a / 



of A must be changed; hence c=o~o, V = ^jt" X ^^r^ — R^, 



^=RxV-^^ 



(or ■ 

 I will now suppose that q3r=any function of r, and F= — , to de- 

 termine the motion ; supposing the particle commences its motion at 

 right 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=R4-a?, 



d:pR 



■a: being always very small; put -Tp"=9'R5 then by Taylor's theo- 

 rem, or by (n) Vol. XX, p. 71, (pr=(p(R-{-x)=(pK-{-x^'K, neglect- 

 ing terms of the orders x^, x^, etc. I also have (as heretofore,) 



_ r^dv . ... 



at=—^ (14), ^=the time, i)=the angle described by the radius 



c ■ (.r 



vector (r) around the centre of force in the time (?), -^=thearea 



described by the radius vector in a unit of time ; I shall suppose that 

 t and V commence when the particle is at the extremity of R, or at 

 the origin. Let V'=the velocity of a particle of matter, describing 

 a circle around the centre of force at the distance R, V=the ve- 

 locity in the curve at the origin, (or at the distance R ;) then (at the 



(pR V'-^ c'"-' 



origin,) F=^=-^, or (pR=^R'V ; R^=^'' °^ c''='R"V^ ; 



(pR V'^ / Vn R?'R 



.*."~7^ = Tr2, put Rl 1 — yF ) =e' and — ^ = m'^ ; then e' is a very 



small quantity because the described curve differs very little ffom a 



circle (rad.=R.) Let -^^the angle at which r cuts the described 



dr 

 curve, then (since -j- =cot. 4',) I have by (H), Vol. XVL p. 286, 



&^ 2&^ cot.24, ■ c'2 /dr^\ 

 F=-r+ --, ~^^[d7^j' °^ ^y neglecting the term 



Jdr 



