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



has an initial velocity of projection, r 3 , the velocity acquired, 

 v, is governed by the following law : — 



: 2 = ^ 3 2 + 



2e(i-J 2 ) • .... (9) 



The path pursued by the body with reference to the centre 



of mass chosen (the centre of the larger body) will be an 



ellipse (or its limiting case a circle), a parabola (which marks 



the boundary between the ellipse and the hyperbola), or 



hyperbola (or its limiting case a straight line), according as 



2e 

 v & 2 is less than, equal to, or greater than — . The increase 



of velocity caused by the approach of the body towards the 

 centre from any distance is independent of the initial velocity 

 of the body, or of the path pursued between s x and s 2 , but 

 depends solely on the magnitudes Sy and s 2 , and is given by 

 the expression 



vW= 2e(i-A) (10) 



10. Assuming that the body is initially at rest at an infinite 

 distance from the centre of mass, equation 10 takes the form 



* 2 =^. (ii) 



If s x becomes zero, then the velocity acquired would be 

 infinite. The distance between the centres of the bodies 

 could not become zero. It is probable that the speed gained 

 by the bodies as they approached would cause Newton's law 

 to be disobeyed, and that even the surfaces of two ordinary 

 particles could not come into actual contact with each other. 

 If s 2 = 2s 1? then equation 10 gives 



*" 2 =7 (12) 



That is, a body acquires as much energy in falling from 2.^ 

 to s 1 as in falling from infinity to 2^. 



11. The energy abstracted from the sether is proportional 

 to the square of the velocity which the body gains due to the 

 action of the attractive forces, and is equal to one-half the 

 mass of the body times the square of the velocity gained. 

 It is perhaps hardly necessary to point out that if the body 

 starts at an infinite distance from the centre of mass it cannot 

 follow an elliptical or circular orbit in approaching, or with 



