Central Forces. 6d 



/ e^ e- 



ing quantities ot the order 6% &c. dv'— \l -\--^cos. ^nt-\--^ 



(cos. wt — COS. 2m,i) Xndt, or by integration v'—nt-{--j^ sin. 2nt-{- 



e^ I sin. 3nz; \ . sin. ^nt 4 . 



— [sin.w^ — ^ P but sin. w^ — — -^ — =-:^sm.^nt; .'.v'=nt 



+ — sin. 2ni;-}--i7-sin.2w^ (20). Put ap'=c^, then, since p' = 



a(l - e"), C" =a-(l — 6^), or c=a ( 1 —-^ I neglecting quantities 

 of the order e', e% &,c. or 2c=a + a(l — e-)=a+/y; hence by put- 

 tinge— p'==D, I have « — c=D, a^ — c- =a"e" =(a+c)D, .'.— = 



(a4-c)D 2 2(a+c)eD 4eD , ^ (a+c)D . . 



^^ — —- — 5 o e =^ = — ^r~o =t; — nearly. Let — r— ; — = sin. Y = 



4a^ 3 3«2 3a ^ 4a2 



4eD 

 Y nearly ,-7r— =sin. Z=Z nearly, (Y and Z being small angles;) Y 



sin. 2?ii=V, Zsin.3?i^— X; then (20) becomes v'=w?;+X+V, 

 which agrees with the second method of approximation given in the 



L 



scholium;- Newton's ~^=p', DO=c, AO=«, PHB=t)^ PSB=t;, 



dv 

 and the angles X, V, are the same as his X, V. Again by (4), —ri 



* _3 14-ecos. t; 



= (1+6003.1?)^ X(l -e=^) ^ also by (2) and (5), — 1_^^ -~= 



1 dv (1 — e^)^ , . 7 / 



; ; hence —rr=7'^ ^rra^id, since na?;=(l-e cos.©) 



1— ecos. 9 ' ndt {l—ecos-cpy \ r/ 



( dcp \- 1 dv I ^V \^ /■ N - / N 



^^' l^j =(r^c'^F'*'^^U^/ X(l-e = )^ (21). 

 I will now investigate a general formula, which will enable me to 

 find V in terms of nt by (21). LfCt Q,=Fy= any function of y, and 

 y= any function of x; [x being a small variable quantity;) then is Q 

 a function of x, which can generally be expressed by a series of the 

 form Q=Q'4-a?Q,+a:=Q,+Jc=^Q3...-j-a?"Q,,+a^''+iQ„+, + etc. (a); 

 Q'j Q,? Q25 &;c. being independent of x. Put a;=0, then (a) be- 

 comes Q = Q', .• . Q'^: the value of Q when x=0', and by taking the 

 differential of (a) n times relatively to x, (supposing dx constant,) I 



d'Q 



have -^=l,2.3...7?Q„-f-2.3...(w+l):cQ„4.,-retc. [b); by putting 



