INTRODUCTION. xv 



arrangement, the orbit of Saturn is affected only by the difference of the perturba- 

 tions by Jupiter and Uranus ; whereas, if the mean places of the perihelia of these 

 two planets were the same, instead of differing by 180°, the orbit of Saturn would 

 be affected by the sum of their disturbing forces. But notwithstanding this favor- 

 ing condition, tbc elements of Saturn's orbit would be subject to very great pertur- 

 bations from the superior action of Jupiter, were it not for the comparatively rapid 

 motion of its perihelion ; its equilibrium being maintained by the very act of per- 

 turbation. Indeed, the stability of Saturn's orbit depends to a very great extent 

 upon the rapidly varying positions of its transverse axis. For, if the motions of 

 the perihelia of Jupiter and Saturn were very nearly the same, the action of Jupiter 

 on the eccentricity of Saturn's orbit would be at its maximum value during very 

 long periods of time, and thereby produce great and permanent changes in the 

 value of that element. But, in the existing conditions, the rapid motion of Saturn's 

 orbit prevents such an accumulation of perturbation, and any increase of eccen- 

 tricity is soon changed into a corresponding diminution. The same remark is also 

 applicable to the perturbations of the forms of the orbits of Jupiter and Uranus 

 by the disturbing action of Saturn; for the secular variations of Jupiter's orbit 

 depend almost entirely upon the influence of Saturn, because the planet Neptune 

 is too remote to produce much disturbance, and the mean disturbing influence of 

 Uranus on the eccentricity of Jupiter's orbit is identically equal to nothing, by 

 reason of the relation which always exists between the perihelia of their orbits. 

 We may here observe that the eccentricity of the orbit of Saturn always increases, 

 while that of Jupiter diminishes, and vice versa. 



The consequences which result from the mutual relations which always exist 

 between the nodes of Jupiter and Saturn, on the invariable plane of the planetary 

 system, are no less interesting or remarkable with respect to the position of the 

 orbit of Uranus than those which result from the permanent relation between the 

 perihelia of Jupiter and Uranus are with respect to the form of the orbit of Saturn. 

 The mean disturbing force of Saturn on the inclination of the orbit of Uranus is 

 about four times that of Jupiter ; but as these two planets always act on the 

 inclinations in opposite directions, it follows that the joint action of the two 

 planets is equivalent to the action of a single planet at the distance of Saturn 

 and having about three-fourths of his mass ; so that the orbit of Uranus might 

 attain a considerable inclination from the superior action of Saturn if allowed to 

 accumulate during the lapse of an unlimited time, at its maximum rate of variation 

 depending on the action of this planet. But such an accumulation of perturbation 

 is rendered forever impossible by reason of the comparatively rapid motion of the 

 nodes of Jupiter and Saturn, with respect to that of Uranus, on the invariable 

 plane. By reason of this rapid motion, the secular changes of the inclination of 

 the orbit of Uranus pass through a complete cycle of values in the period of 

 56,300 years. The corresponding cycle of perturbation in the eccentricity of 

 Saturn's orbit is 69,140 years. It is the rapid motion of the orbit with respect 

 to the forces in the one case, and the rapid motion of the forces with respect to 

 the orbit in the other, that gives permanence of form and position to the orbits of 

 Saturn and Uranus. 



