xviii INTRODUCTION. 



Since the mean motions and mean distances of the planets are invariable, and 

 independent of the eccentricities of the orbits, it would seem that there could be 

 no connection between these elements, by means of which the stability of the sys- 

 tem might be secured or impaired ; but a more careful examination shows that, 

 although the mean motions or times of revolution of the planets are invariable, 

 their actual velocities, or the variation of their mean velocities, depends wholly on 

 the eccentricities ; and, were any of the planetary orbits to become extremely ellip- 

 tical, the velocity of the planet would be subject to great variations of velocity, — 

 moving with very great rapidity when in perihelion, and with extreme slowness 

 when in the neighborhood of its aphelion ; and it is evident that when the planet 

 was in this latter position a small foreign force acting upon it might so change its 

 velocity as to completely change the elements of its orbit, by causing it to fall upon 

 the sun or fly off into remoter space. A system of bodies moving in very eccen- 

 trical orbits is, therefore, one of manifest instability; and if it can also be shown 

 that a system of bodies moving in circular orbits is one of unstable equilibrium, it 

 would seem that, between the two supposed conditions, a system might exist which 

 should possess a greater degree of stability than either. The idea is thus suggested 

 of the existence of a system of bodies in which the masses of the different bodies 

 are so adjusted to their mean distances as to insure to the system a greater degree 

 of permanence than would be possible by any other distribution of masses. The 

 mathematical expression of a criterion for such distribution of masses has not yet 

 been fully developed; and the preceding illustrations have been introduced here, 

 more for the purpose of calling the attention of mathematicians and astronomers 

 to this interesting problem than for any certain light we have yet been able to 

 obtain in regard to its solution. 



