Central Forces. 71 
dQ _,1.,,42Q 
aq & Es) aq ¢@ (r a) dQ’ 
¥ te ae. 3 hence generally da” = asad and Th" 
ony Perhuacd 
ef ( ae aq. dQ 
da (is *, du being the values of r, Ty When 2=0. 
Ce ae dQ’ 
Let n=1, 2, 3, and so on successively in (2); then de da’ 
dy ay 
dQ dr 4 dQ’ a(v al | Lg 
eta? “aos Age? and so on; by substituting 
acy 
dx? ~ 
dQ’ d? d°Q/ gi 
these values of a it oes &ce. in(c) ithecomes Q-Q’+e. (1) 
2? a(n) 5 a? (2) 
eee! = oiviete 
1.2 i Tes ua t ete. (%); put r=0 in (d), 
then y= Lu, and Fy=F(1lu)=Q’; this value of Q’, when substituted 
In (/) gives the general formula which I proposed to find; (see Mec. 
Cel. Vol. I, p.173.) If L=1, ory=u-+ar, then Q=Fy, Q’=Fu, 
dQ’ d 
r= us the value of ‘fy when x=0, and = 3 hence (£) becomes 
% atees dFu et dFu 
= dFu\.. 2 ( “du F ( iz) 
aes. lees ) +73" ac tis ee 
: dFu du 
sa (2); if F=1, or Fy=y, then Q’/=Fu=u, and i gt I coe ; 
2 42 3 2/3 
hence (1) becomes y=utea.r yoiieal eae etc. (m); 
(see Mec, Anal. Vol. II, p. 22.) If fy=1, r’=1, and y=utafy 
becomes y=u+a, and Fy=F(u+2); hence (1) becomes F(u+<2) 
aFu « d?Fu «x? d3Fu \ dis Suen’ 
kis ges See ast ee (n)s t is formula is 
usually known by the name of Taylor’s theorem, and it may also be 
observed that (c) is usually called Maclaurin’s theorem; (see La 
tox’s Traité élémentaire de Calcul Différentiel, ete. pp. 27, 29.) 
Now by (6) Ihave o=nt-Le sin. , which agrees with y=u-+-afy by 
: Fant / 
making Y¥=9, fy= sin. p, u=nt, e=e; put oe =F'’nt, then Fu= 
F dFu 3 es P 
nt, and i =Fnt, r’ =fu== sin. nt; by substituting these values 
