Central Forces, 291 



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

 (Continued from p. 73 of this Volume.) 



Put F(p=l — ecos. <p, then Fw^=l — ecos. n^, ¥'nt=esm.nt ; 



. e^ dsin.^nt 

 hence by (0) 1 -ecos. (p = l — ecos. wz^+e^ sin. - Mi -{--j- 2 'Jt + 



e* d'^ sin.^?zz; r 



2 2 3 — ^dt^ — +etc. but by (5) 1— ecos. (p=-j hence by substi- 

 tuting this value, and the values of sin.^/?i, sm.'^nt, h.c. (see La 

 Croix's Traite de Calcul DifFerentiel, etc. p. 314,) then taking the 

 differentials (as indicated by the formula, making ndt constant,) and 



1 , , . r e^ ^^ ^ e= 



there results the equation -= 1 -f -^ — e cos. 7it — -^cos. znt — 



^ XT- 2 --^-.- g-"^--""" 1.2.22 



e* 

 (3 COS. 3nt — 3 cos. nt) — -i Qcy^a ("*" ^^^- ^'^^- —4.2^ cos. 2nt) — etc. 



(1), (seeMec. Cel. Vol. I, p. 179.) It may be well to observe, that («) 

 gives the solution of Kepler's Problem, supposing v to be calculated to 

 terms, including e" only ; but it is easy to see that the value of v can 

 be easily calculated by the method which I have given to terms in- 

 volving any integral positive powers of e which may be desired. If 

 e>l, the conic section is an hyperbola, a= its semitransverse axis, 

 e= its focal distance ~a, as in the case of the ellipse. In this 



, . «(e^ — 1) p'^dv 



curve p =a[e'^ — l),r=-7—, {2),c'dt=r^dv^=-f-r-, r-; 



^ ^ ^' 1+ecos. ?; ^ ^' (l+ecos.v)2 



V^^ Z7=g^^ —^=n,\henndt=-~-, ^^ (3). By (2) 



a-\-r\" 



— e- 



ae{e^ — 1) sin. vdv a'/e^ - 1 X 



^^'— — /-i I .^^o ..vi — snd sin. u= ? 



(1+ecos. zj)" er 



vdr ct "4" ?' e 

 hence (3) becomes ndt= : — r.=:_t nut = j 



I e \ do 



or r=a \ ~1 (4), then ndt=edtRn.cp ~ — ■ — 5 or by inte- 



\COS. (? I , ' COS. (p ^ 



gration nt=e tan, cp — k.L tan. ( 2+ 4 ) (^)j ^3 9? and t being reckoned 

 from the perihelion, and P— the semicircumference of a circle rad. 



