Temperature and Molecular Attraction. 105 



the distance 2a. The velocity acquired by such motion is 

 given by equation 11 as v 2 = — , and the energy acquired by 

 the body and surrendered to some third body before it could 



DIG 



assume its stable elliptical orbit is -~ — . But exactly this 



£a 



amount of energy must be abstracted from a body in order 

 that it shall move in a circular orbit of radius a. We may 

 therefore state : 



A body moving in an elliptical orbit under the action oj 

 gravitational attraction can be made to move in a circular orbit 

 ivhose radius is equal to the major semi-axis of the elliptical 

 orbit without adding to, or subtracting from, the kinetic energy 

 of the body. 



16. The limited linear orbit is a special case of the ellipse, 

 just as the infinite line was a special case of the hyperbola. 



The total length of the orbit is 4a, the period is — y^ , and 



therefore the average velocity (as regards time) in the orbit 



is —\/—» The actual velocity has varying values from 



zero to infinity. The body in failing from infinity to the 

 distance 2a from the centre of mass must have received, 



according to equation 11, a velocity such that v 2 = — . Now 



in the linear orbit, conceived as commencing at 2a and ex- 

 tending to the centre and then returning, its velocity at the 

 point 2a is zero. By equation 12 the body will have regained 

 as much velocity in falling from 2a to a as it gave out in 

 falling from infinity to 2a when it was brought to rest at 2a. 

 Clearly, therefore, we can substitute for a linear orbit a circular 

 orbit of radius equal to one-fourth of the total linear orbit 

 without changing the energy conditions of the body. 



17. The arithmetical mean distance of a body from the 

 centre of mass, when the body is moving in a linear orbit 

 whose origin is distant 2a from the centre of mass, is a. 

 Similarly when the body is moving in an elliptical orbit the 

 arithmetical mean distance from the centre of mass is the 

 major semi-axis a. Therefore we can state as the result of 

 the discussions under 14, 15, and 16, that 



The energy given out by any tico bodies originally at rest at 

 an infinite distance apart in forming any stable configuration 

 under the action of gravitational attraction is equal to the kinetic 

 energy which they retain and is, for either body, inversely 



