260 Variations of the Arbitrary Constants in Elliptic Motion. 



r dr 



By (49) 1 — e cos. u= -> which gives e sin. tiJw = — > and (51) 





aneft . rdr • . r 2 </r 2 



gives rf« = — > .'.sin. m= ^^' (87), and a 2 e 2 sm. 2 u= a3n3rffa > 



also a(l— ecos. «) =r gives a 2 e 2 cos. 8 m= (a— r) 2 , .'. since sin. 2 



r 2 dr 3 o 2 (l-e 2 ) 

 «+cos. 2 u = l, wegeta 2 e 2 =(a-r) 2 + a2wg(/<a > or - 



2rd*r+ dr a 

 rdr2 u i* >i • a*{l-* 2 Y a a n» dF + \ 



2a ~ r - ^rf^ 2 "' whose dlffer * glves — 7* — = ~ Tfr^ - ' 



r 3 dv 



since 



M= 1, we have by (65) and (70) -^- = ^0(1 -e 2 ), 



2rd 2 r+ dr* 



dv flSv^i — e 2 a 2 n 2 tf* 2 ~~'~ 1 <fo 

 n 2 a 3 = l, .'.—7- = > hence ; =-j->(88). 



for fndt 



<$n'+<5E — fcr-f sin. ttfo a a/ 



fa= -_ J CQS " = -(<S n '+<5E-<kr-|- sin.«3e)^-^n / + 



*E-te+ ^^), (89) ; the variation of (49) or (56) gives br-- 



r dr 



(1 — ecos. u)Sa - acos.u<te-f aesin. u6u= ~6a+ --r (6n' -{- <5E - 



(1 — e 2 )\ <fe r r dr __ . / rdv 



/ a 2 (l-e 2 )\ 8e r dr I 



fa) + ^ _ - r 1 X - = - <5a+ -j (An'+«E - im) + ^a 



6e 



v^l-e 2 X-> (90), substituting the values of bri, 6E from (B) in 

 (90), we get ^«-Aa+^ \3affn dt dR + 2a/n<fta ^- - 



rrftf \ be 



• l-e'teJ + ^-^^l-e^x-. (C), or 6r 



y/l-e 



X 7 > (D). 



By (58)—^ =l + J±- C tan. 2 " = - / i=^°^f J hence 



3 v ' cos. 2 «--g * M-e u * 2 (l-e)cos. 2 « 



2 



2 



f 



by taking the variation of (58), we get (<St> — <te) - = ^1 — « f • <^+ 



a 





