230 THE OCEAN. [Book X. 



Note. — If the movements of the planets accorded 

 exactly with Kepler's empirical laws, then they 

 would also accord exactly with the theoretical action 

 described by the Propositions in this Book with the 

 relative distances calculated from the centre of the 

 sun ; but those laws require correction, because the 

 direct force of the sun's gravitation must be calculated 

 from the centre of the sun, whereas the revolving 

 force is dependent on the centre of gravity of the 

 nearer half of the sun. This observation applies 

 to all the Propositions in this Book in which the 

 revolving force is involved. 



PEOPOSITION V. 



Theorem. — The squares of the velocities of the orbital 

 motions of the planets are to each other directly 

 as the ratios which the revolving force of the sun's 

 gravitation bears to its direct force in their 

 respective orbits. 



For the ratios are inversely as the distances 

 (Proposition III.), and the velocities, measured along 

 the respective orbits, are such that the squares of the 

 velocities are inversely as the relative distances of 

 the planets from the sun ; therefore the squares of 



