260 Variations of the Arbitrary Constants in Elliptic Motion. 



r . . , dr 



By (49) 1 — e cos. «= -' which gives e sin. udu = — > and (51) 



andt rdr r^dr^ 



gives du = -y-> .•.sin.i/.= ^^-^^.(87),anda=e^sm.='u=^^^^^^, 



also a(l— ecos. m) =r gives a^e^ cos.^ u~ (a—r)^, .'. since sin.^* 



r^fZr^ a-(l-e'') 



w+cos.=^M = l, we get a'^e^- =(« - 0' + a^n"di'' ""^ r "^ 



2rcZ2r+</r2 



rdr"" . a''(l— e^)^ a'^n^dt^ "^ 

 2a— r— „ oj^o ' whose differ'l gives = z — > 



r^dv . 



since M = 1, we have by (65) and (70) —T- = Vo\\-e^), 



^rd^rA-dr^ 

 d.v a-\^i — e^ a^n^dt^ "^ dv 



«^«^=i' ■-■nd-t = — ^^— 'h^"^^— Tr^p— =^r'(^^)- 



Substituting n' (or fndt in (54), then taking the variations we have 



5n' + 5E — ^■sJ+ sin. w.(5e a al 



Su= -, = - (Sn'-i-SK — 6zi4- sm. uSe)=-\Sn' 4- 



1 — e cos. u r ^ ' ' r \ 



rdru€ \ 

 5E-«5*+ 2 fit )' ^^^^ ' ^^^ variation of (49) or (56) gives hr= 



r dr 



(1 — ecos. u) Sa - a cos. u8e-\-ae sin. ul)u= -6a+ ~n {^n' + (5E — 



/ a^(l—e'^)\ Se r dr I rdv 



y^l_e2 % — i (90), substituting the values of hn', ^E from (B) in 



r dr I (?R 



(90), we get h = ~ ^«+ ^^ \ ^^ffn dt dR + <^afndt a 2a ~ 



\ / rdv \ be r dr 



^l-e^tej + l^a-^v^l-e^jx-. (C), or^r=-6a-^^ 



dr I dR\ I aH\-e'')\ 



VI -e' Szs-h :;[dt\^^ff^ dt dR+2afndt a ^-j + [a--^- j 



Se 

 X-' (D). 



1 1+e u 1— e cos. u 



Bv ("58^ r— =1+^; tan.2= = 7:i r-j hence 



"j \'^°) cos.^ v-zi ^1—e 2 (1— e)cos.^M 



"2" 2 



by taking the variation of (58), we get {Sv-6zi) - =s/l—e\ i>u-\- 



