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258 Variations of the Arbitrary Constants in Elliptic Motion. - 



u dR 



becomes ede = andl v 1 — e 2 X -r- -a(l — e 3 )dR, (73), which will 



of 





r 2 



Again, since r=a(l — ecos. u), we have (1— c cos. w) 2 = — > 



rfr 



dR 



ae sin. w= tt> d [a(l -e 3 )]=2crfc= - 2r 2 ^ dv by (69) and (65), 

 since M = l, also by (39) da= — 2a 2 dR, and since we neglect quan- 



tities of the order of the square of the disturbing force dR= ~T~dv 



+ 



dR 



dR 



andt 



(1 — \/l—e 2 )dti+2r*du-T-> but du=ndt-±-(l — ecos. u)= f 



dR 



.\ we have dE = (l — v 1— e 3 Jdf*4 %andtr-j-> (74); since r=a 



b 



rfr 



r and then of a, we 



dR , dR r dR , dR dR 



have ^rfr=^x-rfa=- rf - da, .'.r^ =a-^> (75), hence 



(74) becomes c?E=(l - v'l - e 2 )d*+2a 2 ndt j-i (76). 



It is evident by (54) and (58), that » =fndt+ E+<p(fndt+E 

 tf), (77), <p denoting a function of the quantity that follows it ; re- 

 garding R as a function of v and then by (77), as a function offndt, 



dR dR , 

 we have -^ dv = ^f t ndt, (78,) but by (77) dv=ndt [l+<p'(fndt+ 



E — »)], <f>'(Snd/-j-E— «) denoting the differential coefficient of 

 <p(fndt -f E - tf) taken relative to fndt; but by (77) <p'(fndt+'E> 



dv I dv\ ' dR dR 



-,*)*:» hence *=»* I 1 " d^i' ''• < 78 ) becomes ^ - dV 



tf» dR , dRdt; dR , dR dR dR j. 

 X d^=^' or since d^ d^ = te' We have d* = ndt + d^ ' < 79) ; 

 and because that fndt is always accompanied by E in (77), it is ev- 



. dR dR dR dR dR 



ident that ^ = -^ (80), hence (79) becomes -^ = ^ + & ' 



dR 



art 



(81). Substituting the value of -7- from (79) 



dR 



dR 



will be changed to ede = andt v'l -e 2 \^ft + 



