254 . THE OCEAN. [Book X. 



Proof. — Let any planet in any given position be 

 in equilibrium without an orbital motion, and then 

 let the revolving force of the sun's gravitation tend 

 to give it any given velocity of revolution. 



The force of astral gravitation then increases as 

 the square of that velocity (Proposition VIII. ): 



The equilibrium between the opposing forces of 

 solar and astral gravitation is then destroyed, and 

 the planet must therefore move in the direction of 

 the greater force, which consequently carries it farther 

 from the sun : 



As it recedes from the sun, the power of the direct 

 force of the latter to retain it in its orbit decreases as 

 the square of the distance increases : 



But the revolving force decreases as the cube of 

 the distance increases (Proposition lY.) : and the 

 opposing force of astral gravitation decreases in the 

 same proportion : 



Therefore, since the greater force, tending to 

 carry the planet out of its orbit, decreases as the 

 cube of the distance increases ; whilst the lesser force, 

 tending to retain it in its orbit, decreases as the square 

 of the distance increases ; the recession of the planet 

 from the sun must sooner or later restore the equi- 

 librium of the opposing forces. 



Therefore the distance to which the planet is 

 carried from the sun is determined by the action of 

 astral gravitation. 



This demonstration is illustrated by the diagram 



