Variations of the Arbitrary Constants in Elliptic Motion, 259 



dR\ , , dR ^ aV^l-e^ / , \ , ' 



-^)-andt{l-en:;^,'orde=—J—[l-^/l-e^)dR-^ 



«Vl-e2 dR 



Again, since R is a function of fndt, a, E, to', e, p, q, we 



, „ , dR dR dR dR dR 



shall have dR = —7 ?i(/«f = —r. ndt-{- -r- da-\- 7p fZE -f ^ f?ttf -f 



<?R ^ <?R <^R c/R <^R 



Te^'^-dp^P+l^ ^?' (Q4) ; by (72) ^dp+^dq = 0, and 



^ £/R (^R . rfR 



by (80) ^ = —J' also (39) gives Ja= - 2a^ —-judt, substituting 



these values and that of dF, from (76), we easily reduce (84) to 



/ . dR dR\ dR 



^(1_ v/l _e2 ) __|. —j xJ^+ — ^e=0, which gives by (83) 



dR andtVl—e^ dR 

 since rfR= -jndt, cZto'= — X -t-j (85). Substitu- 

 ting the value of d-m in (76), then collecting the results which we 

 have obtained, we have (/a= — 2a^c?R, dn^ = 3fandtdR, rfE= 



andtx^l -e^ , dR dR a\^l-e^ 



, aVl—e^ dR andt\^]-e'' dR 



{l-^^l-e^)dR+ ^ ndt^~,d^=- x-^> 



andt dR andt dR 



It is evident that by neglecting quantities of the order of the 

 square of the disturbing force, we may take the integrals of (A) on 

 the supposition that a, e, &;c. in their right members are invariable, 

 hence by using J for the sign of integration, and (5 for the charac- 

 teristic of variations ; we have (5a= — 2a2/<^R, M=SaffndtdR, 



av/r^ ^ ^ -dR ^ , ^R 



^E=- -—{l-^^l-e^)fndt-^j+2a^fndt^, to= ^ 



a\^l-e^ ^ dR ^ aV\-e^ , . -^ /■ ,„ a^l-e^ 



dR a dR a ^ ^ dR 



fndt-^, ^P=-Vi:rj.f^dtj^, Sq=^f^^fndt^, (B), 



for it is manifest that the integrals of (A) are variations. 



