Central Forces. 71 



d^Q ^\'- 1^) ^ ^, d^Q, ^ \y±j d"Q^ 



^T= j^T- ; hence generally^ = ^^^7:^7 ; and -^ 



= - > „_^ (i) ; r', -1— being the values 01 r, -r- when a;=0. 



Let ^ = 1, 2, 3, and so on successively in {i); then T"'^^''"^' 



^^= d^' -d^= d^ ' and so on 5 by substituting 



dQ'd-'Qd'Q' ^ ^ ^, ( dQf\ 



these vakies of-7-"' -T-v' XT'' ^c. in(c) it beconies Q=Q'+a;. !/•'• ^— 1 



. x^ \ du I . x^ \ du , ,^ • / 7\ 



■^1:2 -du ^"iK^ d^ + ^*^- (^)5 put ^-0 m {d\ 



then y=\u, and Fi/=F(4'm)=Q'; this value of Q', when substituted 

 in iji) gives the general formula which I proposed to find; (see Mec. 

 Cel. Vol. I, p. 173.) If 4. = !, or ?/=M+a?r, then Q=F?/, Q'=Fm, 



JQ' dYu 

 r'=/M=the value of^y when a;=0, and -7— =-r-; hence (A;) becomes 



Yy=Yu\-x. W' 



etc. (Z); if F=:l, or Fij==y, then Q' = Fm=m, and -^="^=1; 



hence (/) becomes 2/='^'+a?-?-'-f j^-^- +iX3"^m^"^ ^^^' W ? 

 (see Mec. Anal. Vol. II, p. 22.) M fy=l, r'=l, and y=u-{-xfy 

 becomes y=^u-\-x, and F?,'=F(m-|-'^); hence (l) becomes F{u-\-x) 

 ^ dFu X' d'-Fu x^ d'Fu r . , r 



=^"+^*~rfT+r:2-^!^+T:2:3* A^+"^"- (")' '^'' ^°'^'^"^" '^ 



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 



Croix's Traite elementaire de Calcul DifFerentiel, etc. pp. 27, 29.) 



Now by (6) I have (p=-7it-{-e sin. v-p, which agrees with y=u-\-xfy by 



dFnt 

 making y~(p,fy= sin. q>, u=nt, x—e; put —t- =--F'nt, then Fu= 



Fni^, and -J— =:F'77/, r'—fus=sin.nt; by substituting these values 



