Chap. XXI.] THE HEAVENS. 243 



Proof. — For the endeavour to carry Neptune 

 round in the same period as Mercury would give 

 the former a velocity of motion 7 7' 7 times greater 

 than the latter. But the revolving force being 

 inversely as the cube of the distance, and therefore 

 469000 times greater in the orbit of Mercury, and 

 increasing its velocity as the square root of the force, 

 or diminishing the relative velocity of Neptune in 

 that proportion, would therefore decrease the relative 

 velocity of the latter 684 times, making the relative 

 velocities as 1 is to ^; or as 8*8 is to 1 ; which are 

 the actual relative velocities of motion along the 

 respective orbits. 



Note. — Books II. to YIII. have shown that the 

 drops of water in the ocean, instead of keeping 

 pace equably with the earth's rotation, tend to 

 lag more at the surface of the ocean in the equa- 

 torial regions than below the surface where nearer 

 the earth's centre ; and the above Proposition shows 

 that the relative lagging of the planets in relation 

 to the sun's rotation accords mathematically with 

 the action of the forces described as circulating the 

 ocean. 



R 2 



