ee SS ae ee ee 
ea Ee eee eS ee ee 
Central Forces. 73 
a) ll 17 ; 13 129 
cos. nt+ ( 5e*—Te! + 96°" ) X cos. 2nt-+ (Fe — ae) cos. dnt 
103 451 1097 1223 
+ yee 120°° Joos. 4nt + J92 ©” ©9S- dnt + 160 & COS Ont; 
multiplying by nd¢ and then taking the integral, I have v=nt+ 
ef 6 5 1 13 
(26 z+ 56°") sin. nt + ( 4° ~aqe" re sin. ant-+( joe*— 
ee toes apr 5 1097 1223 | 
64° sin. Snt+ (gee — 480° sin. 4nt-+- 960 © sin, dnt+ 960° 
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 which 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=ndt x 
2 
é e? _3 ; 
(1454-2¢c0s.04% cos. 20 | X(1—e?) ? (t); if e=0, v=nt, 
and cos. y= cos. nt, this value of cos. v when substituted in (t) gives, 
by neglecting quantities of the orders e?, e°, &c. dv=ndi x (1+2e 
Cos. nt), or by integration v=nt+2esin. nt; this value of v when 
substituted in cos. v, cos. 2v, by rejecting quantities of the orders e*, 
5 . 
€*, &c. gives dv=ndt x ( 1+-2¢ cos. nt-+-5e? cos. ont | and by inte- 
‘ 5 ate : 
gration v=nt+e sin. nt-+-3e* sin. 2nt; by substituting this value of » 
in Cos. v, cos.2v, and neglecting terms which involve e*, e°, &c. I 
e3 i. 
have v=nt+ ( 2e—F} sin. nt+-3e° sin. Qnt+-F5e° sin. 8nt, 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. 
Ste kkeeNo. 1. 10 
