256 Variations of the Arbitrary Constants in Elliptic Motion. 

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



fndt+Yi—'a=u — esm.u, (55) 



r 



(36) 



we have r= r-: — s r 1 — v (57), where we shall suppose E, *r, 



1 + ecos. (y— &) v /7 



and t? 5 to be reckoned from the axis of x in the direction of the 

 motion. 



By comparing (56) and (57), we have tan. ^L- = \/- — 



2 w 1-e 



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



dv A /l+c 



(58), --^^IZ-e 



cos. 2 — 2 



du 



cos. 2 5 



(58) 



u 



2 tan. ~ de 



(59) 



v—a u 1_e (l-c)v / l-e 1> 



cos. 2 — 2 — cos. 2 q 



1 1+e « 



(60) j by (58) -— =1+ fTe tan - 3 2' hence (60) 



cos. 



2 



2 



reduced to (1— e 2 )(J't* -j-sin.tufe +v / l^e 3 x(l— ecos. tt)cfe=0, 



(61). 



Eliminating d'u from (52) and (53), we get (1 — ecos. u) 2 

 da+ae sin. ud (E— ■#)-}- a(e - cos. t*)de=0, also eliminating rf'ti from 



(52) and (61), we have ae sin. u */\ — e 2 x^+a (e + cos. t*) rfe — 

 e 2 )da =0; by adding these equations, we get rfE = (1 - 



/. i, rf[a(l-e 3 )]-(l— ecos. w) 2 rfa ■ ,. t .„ 



vi-e')d*+- k - 1 ^— \ 1 , (62), which will 



aesm. u 



poch 



r 



T /i i " = ^ anc ' (^!) become x=r' cos. v, 1/ = ^ sin. tr, 



, //>«v , xdy — ydx r"dv 



z^s, (63), hence c= - iL ^— = -^-> (64), which by taking 



r 2 dv 

 the plane of x } y for the primitive orbit of m, becomes 7, 





