72 Central Forces. 
2 d(F’nt. sin. id 
in (l) it becomes Fo=Fnt+e(F’nt. sin. nt) +75 
e® d?(F’nt. sin.?nt) 
+ ete. (0); which agrees with (q) given at 
1.2.3 nas" 
page 177, Vol. I, Mec. Cel. Let F=1, then Fo= Ze Fnt=nt, and 
2 
F’nt=1; hence (0) becomes 9 = nt +e sin. nt he 12 ae 
e? d?(sin.®nt) 
1.2.3 n? dt? 
(m), by changing y into 9, u into nt, x into e, and 7” into sin. nt. 
-++ etc. (p); this formula is found immediately by 
By taking the differential of (p) I have . 1+-e cos. m+T 3 
d?(sin.2nt) e® ~+d3(sin.?nt) 
nde +123 n> dt3 
sin.?nt, sin.snt, &c. their values, (see La Croix’s Traité de Calcul 
Différentiel, etc. p. 314,) taking the differentials, (as indicated by the 
formula, making nd? constant,) and by rejecting those terms which 
+ ete. (q); substituting in (q) for 
; or dy 
involve powers of e higher than e®, I have ale cos. ni-+e? 
’ e? e* 3 
cos. 2nt +> g (9 cos. dnt — cos. nt) + 3 (4 cos. 4nt — cos. 2nt) +354 
(625 cos. dnt — 243 cos. Snt 4-2 cos. nt) +7 7 x ~ (243 cos. 6nt — 128 
d 
cos. 4nt-+-5cos. 2nt) (r). It may be observed that oa can also be 
found immediately by the value of u, (given in the Mecanique Ce- 
leste, Vol. I, p. 179,) viz. by taking the rater and then dividing 
both sides of the equation by ndt, and —7, _ will be found to be the 
do 
same as the value of — given by (r), as it evidently pig to be, for 
2 
his u denotes the same thing as g. Now : (21) cae =(%,) x 
a —e8)*; hence by taking the square ped — —~» as given by (r), neg- 
t 
lecting those terms which involve e7, e*, &c. then multiplying by 
(1 ~e)?, (observing that the terms re involve e7, e%, &c. are 
. | oa 
to be neglected as before,) and I have tor | Deu: +96" 
