240 THE OCEAN. [Book X. 



IV.) ; and the direct force, which is inversely as the 

 square of the distance, is a retarding force (Proposi- 

 tion X.) 



Note. — The sun's revolving force, endeavouring to 

 carry the planets round with the same motion, is 

 inversely as the cube of the distance from the sun 

 (Proposition lY.), and the planets are revolved with 

 such relative velocities as to make the squares of those 

 velocities of revolution identical with the relative 

 amount of the sun's revolving force in their respec- 

 tive orbits (Proposition YI.) But the fraction of the 

 sun's direct force iu each orbit which acts as a revolv- 

 ing force is inversely as the distance from the sun 

 (Proposition TIL, Corollary), and the squares of the 

 relative velocities with which the planets move along 

 their orbits are identical with the respective fractions 

 of the sun's direct force which act as revolving forces 

 along the orbits (Proposition Y., Note). And these 

 fractions of the sun's direct force represent its motive 

 action along the orbits of the planets, because the 

 increased force of any given fraction, consequent on a 

 decrease of distance from the sun, is to the same 

 extent counterbalanced in consequence of the direct 

 force forming a part of the retarding force of gravita- 

 tion, which resists the onward motion along the orbit. 



Corollary 1 .■ — If the planets are moved along their 

 orbits in equilibrium between opposing forces of 



