104 Dr. J. E. Mills on the Relation of 



where a denotes the radius of: a circular orbit, or the major 

 semi-axis of an elliptical orbit, or one-half the distance from 

 the centre to the point of return on the linear orbit (one- 

 fourth of the total orbit). 



14. Since the circumference of a circular orbit is 2ira, and 



the velocity = ; — , , it follows that for a bodv 



J period " 



moving in a circular orbit 



27rav€ 

 2ira^ 



= \/~ and v*=- . . . (14) 

 V a a 



Now the velocity squared, acquired by the same body in 

 falling from an infinite distance to its circular orbit is given 

 by equation 11 and is 2e/a. Evidently, therefore, one-half 

 of the total energy acquired by the body must have been 

 transferred to some third body before it could assume its 

 stable circular orbit, and we can state : 



Two bodies moving in circular orbits wound their common 

 'centre of mass under the action of gravitation before they could 

 assume this position of stable equilibrium must have surrendered, 

 to some outside third body one-half of the kinetic energy which 

 they gained from the cether by their approach. 



15. The speed of a body moving in an ellipse is at every 

 point equal to that which it would acquire in falling from 

 the circumference of a circle, with centre at the origin and 

 radius equal to the major axis of the conic, to the ellipse. 

 For the velocity when moving in an elliptical orbit is given 



©■ + (l)"=--(!-D- • ■ < 15 > 



where S] is the distance from the origin and a is the major 

 semi-axis, From equation 10, making the velocity in the 

 orbit equal the velocity acquired in falling from s 2 , we have 



.■(*-iV=2 e(L-I) (10) 



\s 1 a J \si sj 



Therefore the energy retained by a body moving in an 

 elliptical orbit is equal to the energy that it would acquire 

 in falling into its orbit from a distance from its centre of 

 mass equal to the major axis, 2a, of its orbit. The body 

 must have previously surrendered to some third body an 

 amount of energy equal to that which it would acquire in 

 moving from an infinite distance from the centre of mass to 



