SECULAR VARIATIONS OF THE PLANETARY ORBITS. 2G9 



The perihelia and nodes of their orbits liave the following- mean mo- 

 tions in a Jnlian year of 365:^ days : 



Mean motiou of pcrihi'Iion, Mean motion of node on the 



invarialde plane. 



Jupiter + S^.TIGGO" — 25".931."iG7 



Saturn +22 .4G0848 — 25 .9345G7 



Uranus + 3 .71GG07 — 2 .91G082 



Keptune + .G1GGS5 — .GGlGGo 



But the most curious relation developed by this investigation per- 

 tains to the relative motions and positions of the perihelia and nodes of 

 the three planets Jupiter, Saturn, and Uranus. These relations are ex- 

 pressed by the two following theorems : 



I. The onean mol 1071 of JH2)itcy''s perihelion is cxactJij equal to the mean 

 motion of the perihcJion of Uranus, and the mean longitudes of fnese peri- 

 helia differ by exactly ISO"^. II. The mean motion of Ju2)iter''s node on the 

 inrariahle plane is exactly equal to that of Saturn, and the mean longitudes 

 of these nodes differ by exactly 180^. 



We also perceive that the mean motion of Saturn's perihelion is very 

 nearly six times that of Jupiter and Uranus, and this latter quantity is 

 very nearly six times that of x^eptune ; or, more exactly, 085 times the mean 

 motion of Jupiter's perihelion is equal to 1G3 times that of Saturn^ andi-iO 

 times the mean motion of JSTeptune's perihelion is equal to 73 times that of 

 Jupiter and Uranus. The perihelion of Saturn's orbit performs a com- 

 plete revolution in the heavens in 57,700 years; the perihelia of Jupiter 

 and Uranus in 348,700 years; while that of Neptune requires no less 

 that 2,101, 5G0 years to complete the circuit of the heavens. In like 

 manner the nodes of Jupiter and Saturn, on the invariable plane, perform 

 a complete revolution in 49,972 years ; that of Uranus in 444,432 years; 

 while the no<le of Neptune requires 1,958,G92 years to traverse the cir- 

 cumference of the heavens. The motions of the nodes are retrograde 

 and those of the perihelia are direct ; thus conforming to the same law 

 of variation as that which obtains in the corresponding elements of the 

 moon's motion. 



TTe may here observe that the law which controls the motions and 

 positions of the perihelia of the orbits of Jupiter and Uranus isoftlie utmost 

 importance in relation to their mutual perturbations of Saturn's orbit. 

 Uoi', in the existing arrangement, the orbit of Saturn is affected on\y by 

 tlie difference of the i)ertur!)ations by Jupiter and Uranus; whereas, if 

 the mean places of the perihelia of these two planets w^ere the same, 

 instead of differing by 180^, the orbit of Saturn would be affected by the 

 sum of their disturbing forces. But notwithstanding- this lavoring con- 

 dition, the elements of Saturn's orl)it would be subject to very g^reat 

 perturbations from the superior action of Jupiter, were it not for the 

 comparatively ra|)id motion of its perihelion ; its equilibrium being main- 

 tained by the very act of perturbation. Indeed, the stabilit}^ of Saturn's 

 orbit depends to a very great extent upon the rapidly varying positions 



