34 



ii] zz h, therefore at the Apsids 



2 _ idz^ ill 



(It ' x7tsty~ (list - 



hence ^i^+>l-~ *'' 



a(l_c) ' ■" n' (I— f) = 



•,-r A; =- 



= 



These two equations give 

 h-'^a (1— e2 )M 



a 



Substituting these values in equation {b) and making P 

 andR-i^ 



|2 I ^ 



x/f "_J 1 



x' V a(l— e') I' 



The integration of this equation (making the longitude of Peri- 

 gee — tt), gives v = arc (cos = -7^— ■- > + -^ 



or cos (v _ rr) :^ — -— — - 



and therefore 



* — l+t' ens {v— i=r) ^ 



This is the equation of an ellipse, the centre of force being in the 



focus. 



The second law of Kepler follows from this conclusion. 



It may be remarked here that the first law of Kepler follows 



from equation (7) when F = o, for it gives z = h t, as Ave may sup- 

 pose z and t to commence together, and therefore the areas 

 about a fixed centre are proportional to the times of describing 

 them. Also by help of equation (16) we deduce a conclusion that 



