74 



KINETICS OF A PARTICLE. 



[141. 



141. The relation of the eccentric angle <j> to the polar co-ordinates 

 r, will appear from Fig. 22, in which P is the position of the planet 



Fig. 22. 



at the time /, P' the corresponding point on the circumscribed circle, 

 ^.AOP the true anomaly, and % ACP'=(f> the eccentric anomaly. 

 The focal equation of the ellipse 



r= _ _L = (t-Y 



i + e cos 6 i + e cos 



gives r + er cos 6 a ae 1 ; and the figure shows that r cos = a cos <J> ae 

 hence, 



r=.a(\ e cos<), or a r=z ae cos <. (28' 



Equating this value of r to that given by the polar equation of the 

 ellipse, we have 



i e cos 



, orcosO= 



i + e cos ' i e cos <f> 



A more symmetrical form can be given to this relation by computing 



i-cos0=2sin 2 -=(i+<?) I - 



i e cos <j> 



i + cos = 2 cos 2 - = (i e) T + cos 9 

 2 i e cos <J> 



whence, by division, tan- = \/^ t-^ 



2 V !_^ 



(29) 



142. To find t in terms of r, we have only to substitute in (24) for 

 iP its value from (8), Art. 113, and to integrate the resulting differential 

 equation 





