Central Forces. 73 



/5 11 17 \ /13 129 \ 



COS. w^+( ^e^— TQ^^+gge^ JXcos.2n<;+ V~^e^ — -q^c^ j cos.2nt 



I 103 451 \ 1097 1223 



"^ I T4 ^' ~ 120^' r^^' ^"^ "^ T92 ^' ^°^* ^"^ + 160"^'^°^" ^"^' 

 multiplying by ndt and then taking the integral, I have v — nt-\- 

 I e^ 5 \ /5 11 17 \ /13 



I ^^~ 4+96'' ] sin-^i^+ [ 4^^ -34^* + 192^'] sm.2n^+ |^ J2'" ~" 

 43 \ / 103 451 \ 1097 1223 



sin. 6nt (s) ; this value of v is the same that La Place has found at 



page 181, Vol. I. of the Mecanique Celeste, and if I am not greatly 



deceived the method w^hich I have used is altogether more simple 



and easy than his. Again v is easily calculated by Newton's method 



of repeated substitutions; for (4) is easily changed to dv=^ndtx 



/ e- e" \ _3 



( 14-2" +2ecos.?; + ^ C0S.2?; I X(l -e") ^ {t); i? e = 0, v = nt, 



and cos. v= cos. nt, this value of cos. v when substituted in [t] gives, 

 by neglecting quantities of the orders e", e^, &;c. dv—ndty.{\-\-2e 

 COS. nt), or by integration v=^nt-\-'2esm.nt; this value of i) when 

 substituted in cos.v, cos. 2?;, by rejecting quantities of the orders e^, 



e*, &ic. gives dv=ndtx ( l+Secos.ni+^e^ cos. 2w^ j and by inte- 



gration v=nt-\-2e sin. nt-}-^e^ s'm.2nt; by substituting this value of v 



in cos. v, cos.2y, and neglecting terms which involve e"*, e^, Sic. I 



/en 5 13 . 



have v=nt-\-\2e — -Tjsm.nt-\'-^'^^sm.2nt-{'T^e^sm.Snt, and by 



repeating the process for the fourth powers of e, then for the fifth 

 powers, and then for the sixth powers the same value of v will be 

 found as given by (s) ; but this method, although very simple for a 

 few of the first terms of the series, becomes ultimately very laborious. 



Vol. XX.— No. 1. 10 



