282 



rQ 



find from simple geometrical considerations ^' cos ^+17, <^^^' = ^. 





 and consecjuentlj 



<P=zOïy -\- G; 4- fh'dr. ..".... (9) 



'0 



Next calling the values of r for which R vanishes a {1 — e) and 

 a (1 -j- e) respectively, we iind 





(10) 



R^ z=z ct^m 



— 1 4- 



2a u-{l—e'y 



I now introduce a new parameter d by 



G 



7 = 



l/m 



We have then 



d0 1 röi? 



•^ dG ^ [/mj dy 



(H) 



(12) 



Putting now 



/=^- /; = 





dR 



we find from (12) and (10) 



dr _ r'R 



df y [/7)i 



4- 



(13) 



a-[l — e') ■ a(l— e^) 



This is the differential equation of an ellipse of which a is the 

 semi major axis and e the excentricity. If the constant of integration 

 is so chosen that r^ := a (1 — e), then ƒ is the true anomaly. We 

 have then 



a {l—e') 



R z= a \/m 



e sin f 



• (14) 



\-\-e cos f 



We can now in this ellipse introduce by definition tlie excentric 

 anomaly e and the mean anomaly m. We find 



r ■= a [l — e cos e) 



R z=i u |/?/i 



ae sin e 



