Variations of the Arbitrary Constants in Elliptic Motion. 255 



quantities of the order of the square of the disturbing force, (suppo- 

 sing the plane of x, y to be taken for that of the primitive orbit of 



m,) b 3 =c'+c"+c"'=c>, .'.;>' = «(l_e 2 )=||> (44), hence 



dcj * , /-, dc . I dR dR\ 



A 



~7=|, (45), which will enable us to find the variation of e ; and it 

 may be observed that the variation of a is found by (39). Since 



i 2 



in2n= 'v 



M 



jyj* =a(l — e 2 ), by putting n = v — or n 2 a 3 =M, (46), we get 



rdr 



3na 



by (41) ndt = ^ ^^ ^ (g-^ ( 47 ) ; < 46 ) S ives dn= M x 



d(M\ 3na 



5.1 jT J or by (39) dn = -*ir dR, and usingy for the sign of integra- 



tion n=J % -*r-dR, put dn f =ndi and we have n'=fndt= ^ffna 



dtdR 9 (48), which gives the variation of the mean motion; sup-* 

 posing fn dt to denote the mean motion of m. By putting a 

 r = aecos.u, or r = a(l — e cos* #), (49); (47) becomes nrff 



(1 — ecos. w) Xflf [a(l — ecos. «)] 



— n ' ■ — > (50) ; since a and 6 belong to 



ae sin. ti % ; // & 



the ellipse in which m is momentarily moving, we have by (50) 



ndt = (1 — 6 cos. u)du, (51), by taking the total differential of 



a(l — e cos. u) in (50), having regard to (51), we get (1 — e cos. ») 

 da — a cos. ude+aesm. ud'u=0, (52), where d'u denotes the dif- 

 ferential of u which arises from the variability of the elements of el- 

 liptic motion.. 



Put d (E — "tf) = (l — e cos. v)d'u— sin. ude, (53), adding this to 

 (51), then taking the integral, we haveyWtf+E — tf=tt-esin. u 

 (54) ; where m= the excentric anomaly reckoned from the perihel- 

 ion of the ellipse, E the longitude of the epoch, and ta the longitude 

 of the perihelion, which are reckoned in the plane of the orbit in 

 the direction of the motion from any straight line drawn at pleasure 

 through the focus where M' is situated. 



Again, by taking the plane of a?, y, for that of the primitive orbit 

 of m, by neglecting quantities of the order of the square of the dis- 

 turbing force, we may suppose that (54) is the orthographic project 



