72 Central Forces. 



in {I) it becomes Yii;>-—¥nt-\-t{Y'nt.s\n.7it)-\-::-^ — ^^ -r— — --\- 



e^ d'^iF'nt. sin. ^nt) 



T273' ^IF' "^ ^^^' ^^^ ' ^^^^*^^ ^^^®®^ ^^^^ (^^ Siven at 



page 177, Vol. I, Mec. Cel. Let F=l, then Fcp=9, Fnt=nt, and 



F'w^ = 1 ; hence (0) becomes cp=-nt-\-e sin. nt 4- z-x'~ — "x — - + 

 ^ ^ 1.2 na^ 



e^ fZ^ fsin.^wO 

 T~o^ — 2*7^3 "I" ^^^- (P) J "^^^^ formula is found immediately by 



(m), by changing y into 0, u into 7it, x into e, and r' into sin. ?i?^. 

 By taking the differential of (p) I have -T- = l+e cos. w^+r-^- 



d^{&m.^nt) e^ d^{s'm.^ni) , . . ■ . ^ r 



"^'^^■^1X3 n^'dt~'^^^^' (^)' substitutmg m {q) for 



sin.^n^, sin.sw^, &c. their values, (see La Croix's Traite de Calcul 

 Differentiel, etc. p. 314,) taking the differentials, (as indicated by the 

 formula, making ndt constant,) and by rejecting those terms which 



involve powers of e higher than e% I have --T, = l+ecos. w^-fe^ 



e^ e^ e^ 



cos. 2nt + "^ (9 cos. ont — cos. nt) -{- -g (4 cos. 4nt — cos. 2nt)-\--^. 



(625 cos. bnt - 243 cos. 3nt -\- 2 cos. nt) 4- j^ (243 cos. ^nt — 128 



cos. 4w^+5cos.2n^) (r). It may be observed that —1 can also be 



found immediately by the value of u, (given in the Mecanique Ce- 

 leste, Vol. I, p. 179,) viz. by taking the differential and then dividing 



du 

 both sides of the equation by ndt, and ^ will be found to be the 



same as the value of -j given by (r), as it evidently ought to be, for 



dv I dp \'^ 

 his u denotes the same thing as (p. Noav by (21) ^^= |^ ^^ j X 



(1 — e^)^; hence by taking the square of ^v as given by (r), neg- 

 lecting those terms which involve e'', e^, he. then multiplying by 



(1 _e2)2^ (observing that the terms which involve e\ e% he. are 



dv j e^ 5 \ 



to be neglected as before,) and I have ^^ = 1+ ( 2e-- ^-t-g^e* I 



