Chap. XXI.] THE UEAVENS 249 



But Neptune lags 683 revolutions whilst Mer- 

 cury lags one ; and the length of its orbit is 77 

 times greater ; so that the velocity of its lagging 

 motion is 52,591 times greater. The square of this 

 increased velocity is 2,765,813,281, which is approxi- 

 mately the fifth power of the distance. 



Note. — In the above (for the reason given in 

 the Note to the foregoing Proposition) the relative 

 periods of revolution are taken as the measure of 

 the relative velocities of the lagging motions, making 

 Neptune's lagging approximately 683 revolutions 

 more than Mercury's, because the latter takes about 

 88 days to revolve and the former 60,126 days. 

 The limit of the correction which may be requisite, 

 and which is not, even at its maximum, enough to 

 affect the present argument, is shown in connection 

 with the foregoing Proposition. 



PROPOSITION XXL 



Theorem. — The relative velocities of absolute motion 

 with which the planets move along then- orbits 

 are to each other directly as the square roots of 

 the fractions of the sun's force of gravitation, 

 which act as revolvmg forces in each orbit respec- 

 tively.^ 



^ This is a corollary to Proposition III. 



