I43-] CENTRAL FORCES. 75 



As, by (20), Art. 135, c i = ^/a = ^a(i e 2 ), this equation becomes 



or 



la rdr 



dt= \/ -- 



VVaV-o- 



The integration is easily performed by introducing the eccentric 

 angle <J> as variable by means of (28) ; this gives 



= \- 



a(i e cos 



If the time be counted from the perihelion passage of the planet, we 

 have /= o when r = a ae, i.e. when $ = o ; hence, putting V/*/tf 3 = n, 

 .as in Art. 139, we find 



nt = <f> e sin <. (30) 



This relation is known as Kepler's equation ; the quantity nt is called 

 the mean anomaly. 



143. Kepler's equation (30) can be derived directly by considering 

 that the ellipse APA (Fig. 22) can be regarded as the projection of 

 the circle AP'A', after turning this circle about A A ' through an angle 

 = cos" 1 (/#). For it follows that the elliptic sector A OP is to the 

 circular sector A OP' as b is to a . Now, for the circular sector we have 



A OP = A CP - O CP' = J flfy - 1 ae - a sin < = - (< - e sin <) ; 

 hence, the elliptic sector described in the time / is 

 AOP=- ' 



a 2 



The sectorial velocity being constant by Kepler's first law, we have 



hence, /= (<- 



27T 



and this agrees with (30) since, by (25), 27r/7 T = . 



