258 Variations of the Arbitrary Constants in Elliptic Motion. 



, cZR 



becomes ede = andt v 1 ~ e^ x -^ - a(l - e^ )fZR, (73), which will 



enable us to find the variation of the excentricity. 



Again, since r=fl(l — e cos. m), we have (1— e zq%.uY = — > 



dr d^ 



acsin.M=^' (7[a(l -.e2)]=2c<;c= - Sr^ -j^dv by (69) and (65), 



since M = l, also by (39) da— — Sa^rfR, and since we neglect quan- 

 tities of the order of the square of the disturbing force d^— ~T~ dv 



ofR 



+ -"t" dr ; by substituting these values in (62), it becomes dYi = 



- cZR andt 



(1 — V 1 — e2 jJ^_|_2;-2 j^ _-, but du=ndt-^{\ — e cos. m)= y 



, dR 



.'. we have r?E = (l — V 1— e^ )j5}'-[-2anJ2;r-T-' (74); smce r=a 



(1 — e COS. m), by considering R as a function of r and then of a, we 



dR c?R r rfR rfR </R 



have^rfr=:^X-^a=^-c/a, .-.r^ =a^, (75), hence 



, rfR 



(74) becomes dY. = {\-V \- e^)d^-^'ia-'^ndtj'^^ (76). 



It is evident by (54) and (58), that v =/7i(?HE4-<p(/wcZ^+E 

 — •33"), (77), 9 denoting a function of the quantity that follows it; re- 

 garding R as a function of ?;and then by (77), as a function o^fndt, 



JR dR 



we have -j- dv=—rndt, (78,) but by (77) dv=ndi[l-{-(p'{fndt+ 



E — •cj')], <p'(Sn£?^+E— ■zs) denoting the differential coefficient of 



q){fndt + 'Ei-T!^) taken relative to fndt; but by (77) (i^'{fndt-{-'E. 



dv ( dv\ dR dR 



-^)=-5^' ^^""^ ^"=^^^ V-d^l' •■• ^^^) becomes^ - ^- 



dv dR dRdv dR dR dR dR 



^ d^=^t' °' ^^""' ^ ^ = ^' ^" ^"^" Ib^^dt + W C^^) ' 



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



dR dR , , , dR dR dR 



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



rfR , . 



(81). Substituting the value of -j^ from (79) in (73), smce dR=^ 



dR , , / (dR 



-^- xndt, (82), it will be changed to ede = andt V I - e^ 1 -j- -{- 



