256 Variations of the Arbitrary Constants in Elliptic Motion. 



tion of the curve described by m, when reduced to the plane of x, 

 y; hence we have J'ndt-\-Fi — 'ui=u — e sin. u, (55), and by (49) 

 r=a(l — e cos. u), (56), also by substituting the value of p' in (36), 



a(l-e2) 



we have r= r~, 7 \' (57), where we shall suppose E, -sr, 



1 + e cos. (w— ■3J) ^ >" rr 5 ' 



and V, to be reckoned from the axis of x in the direction of the 



motion. 



By comparing (56) and (57), we have tan. 



V — T:i . /1 + e 



n/| 



2 " 1-e 



tan. q) (58), since the elements are constant in the ellipse in which 



dv /l + e 



mis momentarily moving, we have by (58), — ~~— ^ TZT 



cos.^ — ^ 



du 



' (59) ; by taking tbe total differential of (58), and noticing 



u 



cos. 2 



2 



u 

 J J/ /T~f — 2 tan. - de 



dvi d'u \/^~'~^ 2 



(59), we get -, = x V -. — - + 7- — , 



cos.^-y- COS.^g ^^ ^)^ ^ ^ 



1 1+e M 



(60) ; by (58) ~~ =1+ j-^-^ tan. 2 ^, hence (60) is easily 



COS.- — 2~ 



reduced to (1— e^) d'u + sin. ude +v 1 — e^ x(l — ecos. m)Jto'=0, 

 (61). Eliminating cZ'w from (52) and (53), we get (1 — e cos. uy 

 da-\-ae sin. ud (E— -33^)4- a(e — cos. u)de=0, also eliminating £?''« from 



(52) and (61), we have ae sin. u v 1 — e- XcZ53'+a(e+ cos. u) de — 



{\—e^)da =0\ by adding these equations, we get c?E = (1 - 



. d\a(\ — e^M — (\— ecos. uY da 



\/l_e2W^_^_L_v Ai__v 1 ^ ^g2), which will 



enable us to find the variation of the longitude of the epoch. 



r 

 Put -7==- = r', and (31) become a;=r' cos. 1;, y = r'sm.v. 

 VI + s^ ' V y , u > 



xdy — ydx r'^dv 

 z—r's, (63), hence c= ^ — = ~Tf^ (64), which by taking 



r'^ dv 

 the plane of x, y for the primitive orbit of m, becomes —77- = 



