Central Forces. 65 



Art. VII. — On Central Forces ; by Prof. Theodore Strong. 

 (Continued from Vol. XIX. p. 49.) 

 p' 



By (5) given at p. 48, Vol. XIX, r^j^j^—^ (1); or, since^ 



a(l — e") , , , , , , p'^dv 



=a(l -e^), r=Tn (2) ; hence c'dt—r^dv=Y^[-, rz 



"'K^ ^ )-> 1+ecos.t?^ '^ (1+ecos. u)2 



3 



(3). Put -r=g-, and V |- = n, then by (3) nc/i^^jqiT^^:^ 



(4) ; nt==- the mean anomaly, t?= the true do. ; which are here 



ae{\ — e^)sin. vdv 

 counted from the perihelion. By (2) dr— d , g(.Qg ^\T~ and 





sin. u= : hence ndt=^ 



(1+ecos. «)2 



rdr a — r 



=. ; — — =- ; put — =e cos. 9 or r=a (1 — e cos. <p) 



/ la-r\'^ « 



(5); then ndt=^{\—e cos. (?) d(p or by integration nt=(p — es\n.^ 

 (6)5 9= the eccentric anomaly, which is supposed to be reckoned 



from the perihelion : by comparing (2) and (5) yT 



l-e- 



e COS. V 



= 1 



e cos 



V /\-\-e (p 



. cp ; hence tan. ^= v r-^ X tan. 5 (7)- 1^ e = l, the conic 



V 2 



section is a parabola; and since l+cos. v=2cos.'^ = 



l+tan.2- 



(1) becomes r= =— Il4-tan.=^^ (S) ; and (3) becomes 



2 COS.- 2 



c'd!^=-— ( l+tan.=^5 I^Ztan.^s or by integration c't=--^ I tan.^-f 



V 



tan.=5- 



2 p'- I V v\ . ^ 



g— ^ ; put c't=%K, then-^ ^3 tan.^ + tan.^^j ==^^ (^)- P"* 



--=/^ and assume Rcos. ^^g-, Rcos. (d — ())—, ^ (10); hence R~ 

 Vol. XX.—No. 1. 9 . 



