140.] CENTRAL FORCES. 



73 



This convenient expression for the velocity in terms of the radius 

 vector might have been derived directly from the fundamental relation 

 (5), v = c/p, the first of the equations (19), (? = //,/, and the geometrical 

 properties of the conic sections (r r' = 2a, pp' = & 2 , p'r = pr', where 

 r, r 1 are the focal radii, and /, /' the perpendiculars let fall from the foci 

 on the tangent) . The proof is left to the student. 



139. Time. In the case of an elliptic orbit, the time T of a complete 

 revolution, usually called the periodic time, is found by remembering 

 that the sectorial velocity is constant and = 1 c (Art. in), whence 



rp _ 



or, by (20), T=2 = . (25) 



The constant n 



which evidently represents the mean angular velocity in one revolution,. 

 is called the mean motion of the planet. It should be noticed that it 

 depends not only on the intensity of the force, but also on the major 

 axis of the orbit, while in the case of a force directly proportional to the 

 distance it is independent of the size of the orbit (see Art. 125, Ex. 7). 

 The periodic time 7"and the major axis a of a planetary orbit deter- 

 mine the intensity p. of the force 



^ = 4- 2 ^, (26)- 



whence F= m f(r) = m^ = 4*?i*jzp (27) 



where m is the mass of the planet. 



140. To find generally the time / in terms of or r, we can, of course, 

 proceed as indicated in Art. 114; but the resulting expressions are 

 somewhat complicated, and it is best to introduce the eccentric angle 

 < of the ellipse as a new variable, and to express /, r, and 6 in terms of 

 <. In astronomy, the polar angle is known as the true anomaly, and 

 the eccentric angle < as the eccentric anomaly. 





