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 



«i,) 62=c2-fc'2-f c''2=c% ,'. p' = a{\-e^) — ^^ (44), hence 



dc^ , _^ / <?R rfR\ 



<??'== M' and f?\/a(l-e2)=^|^= [by (16)], [V]}--^ jy) 



dt 

 -y=, (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 

 |rF=a(l— e^), by putting n= v — or n'^a^=M, (46), we get 



rdr 2na 



by (41) ndt = ^^^^^n-(<ri7)^' (^'^) ' W Sives dn= ^ x 



2 1 — 1 or by (39) dn = -iuF aR, and usingy for the sign of integra- 



tion n=J'-j^dR, put dn'=ndi and we have n'=fndt~ Tjiffna 



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

 posing J ndt to denote the mean motion of m. By putting a — 

 r — ae COS. u, or r = a(l — e cos. w), (49); (47) becomes ndt = 

 (1 — c cos. u)y:d [a(l — e cos. u)] 



ae~snuu ' ^^^^ ' ^'"^® " ^"^ ^ belong to 



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

 ndt = (I — e COS. u)du, (51), by taking the total differential of 

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

 da — a COS. ude-^aes\n.ud'u=0, (52), where df'M denotes the dif- 

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

 liptic motion. 



Put rf(E— trf) = (l— ecos. w)fZ'M— sin. tt£?e, (53), adding this ta 

 (51), then taking the integral, we havey«f?^+E— to-— m — esin. w 

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

 ion of the ellipse, E the longitude of the epoch, and •cr 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 x, 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 projec- 



