Variations of the Arbitrary Constants in Elliptic Motion. 259 



dR 



dvs 



andt(l — e 2 ) 



dR 



ndt 



or de 



a<S\ 



e 



e 



l-VT 



e 



dR + 



ay/\ 



e 



dR 



e 



X j- ndt, (83) 



Again, since R is a function of J 'ndt, a, E, •#, e, p, q, we 





shall have dR 



dR 



dR 



dR 



ndt 

 dR 



ndt 



dR 



ndt 



dR 



dR 



dR 



ndt+ —da+ Jr , dE -f ^7 dm + 



da 



dE 



dm 





dR 



dR 



Te de + ~dp d P+ !£**> ( 84 ) | ^ ( 72 ) di d P+ dj ** 



0, and 



by (80) 



dR 



dE 



dR 



ndt 



also (39) gives da 



dR 



2a 2 —7 ndt, substituting 



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



(1-v/l 



L dR 



dR 



dR 



e ' ) JiA + A7J x<fa+ A rfe=0 > which S ives b 7 (83) 



since dR 



dR 



n 



-t ndt, dtf 



andt \/I 



e 



dR 



e 



X -p (85), Substitu- 



ting the value of efa in (76), then collecting the results which we 



have obtained, we have da 



2a 2 dR, dn'=ZfandtdR, dE 



an 



dtV\ 



e 



e 



(\-V\-e 2 ) 



dR dR 



+ 2a 2 ndt — « de 



de 



da 



a\/\ 

 e 



e 



(l_v/l-e 2 )rfR+ 



aV\ 



e 



e* dR 



an 



dts/\ 



e 



e 



X 



dR 



de 



dp 



andt 



v\ 



dR ■ 



x T q ' d * 



VI 



an dt dR 

 = X -j > (A). 



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 f for the sign of integration, and 6 for the charac- 



teristic of variations ; we have 6a 



2a 2 fdR, 6n'=3affndtdR, 



<5E 



I 



avT 



aV\ 



e 



dR 



dR 



e 



(l-Sl-e^fndt-^+Wfndt-^ 



Sm 



e 



dR 



e 



■J* ndt t:> &e 



ay/l 



e 



de 



e 



Vl-e 2 )/(/R+ 



aV\ 



e 



e 



r J dR « 

 fndt ^ , 6p 



a 



vi 



e 



fndt 



m 



da 



6q 



a 



V\ 



e 



fndt 



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



dR 



dp 



(B), 



