88 



NEWTON S PKINCIPIA. 



and v the velocity in a circle,) as ; and v being as -, 



a a 



the centrifugal force is as -3, or inversely as the cubes 



CL 



of the distances. A C B is the fixed ellipse ; a C b, the 

 one described by the body under the influence of a 

 disturbing force, or in any other way made to move 

 in an orbit whose axis, I S, or line joining the apsides a, 

 b 9 is revolving round S. Suppose the angular motion 



of the second ellipse to be in a given proportion to 

 the motion of the body in the first, or that sp being 



equal to S P, the angle B S P is - of the angle b S p. 



The difference of the centrifugal forces of the two bodies 

 must be equal to the difference of their centripetal 

 forces. Calling T and t the centrifugal forces in the 

 fixed and moveable orbits respectively; C and c the 

 centripetal forces; T t C c, and c = C + t 

 T. But T : t in the proportion of the squares of the 

 velocities, or of the angular motions, that is as m 2 : n 2 

 and T t : t : : m 2 n 2 : n 2 ; and because the centri 

 fugal forces are at different distances, inversely as the 

 cubes of those distances, therefore the difference of those 

 forces in the two orbits, being in a given ratio to either 



