xiv INTRODUCTION. 



Maximum eccentricity. Minimum eccentricity. 



Jupiter 0.0608274 ' . . . 0.0254928 



Saturn 0.0843289 0.0123719 



Uranus 0.0779652 0.0111576 



Neptune 0.0145066 0.0055729 



The maximum and minimum inclinations of their orbits to the invariable plane . 

 of the planetary system have the following values: — 



Maximum inclination. Minimum inclination. 



Jupiter 0° 28' 56" 0° 14' 23" 



Saturn 1 39 47 16 



Uranus 1 7 10 54 25 



Neptune 47 21 33 43 



The perihelia and nodes of their orbits have the following mean motions in a 

 Julian year of 365 J days: — 



Mean motion of perihelion. Mean motion of node on the 



invariable plane. 



Jupiter + 3". 716607 — 25".934567 



Saturn +22.460848 ■ . —25.934567 



Uranus +3 .716607 — 2 .916082 



Neptune +0 .616685 — .061666 



But the most curious relation developed by this investigation pertains to the 

 relative motions and positions of the perihelia and nodes of the three planets, 

 Jupiter, Saturn, and Uranus. These relations are expressed by the two following 

 theorems : — 



I. The mean motion of Jupiter 's perihelion is exactly equal iorthe mean motion of 

 the perihelion of Uranus, and the mean longitudes of these perihelia differ by exactly 

 180°. II. The mean motion of Jupiter's node on the invariable 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 Neptune; or, more exactly, 985 times the mean motion of Jupiter's peri- 

 helion are equal to 163 times that of Saturn, and 440 times the mean motion of 

 Neptune's perihelion are equal to 73 times that of Jupiter and Uranus. The peri- 

 helion of Saturn's orbit performs a complete revolution in the heavens in 57,700 

 years; the perihelia of Jupiter and Uranus in 348,700 years; while that of Nep- 

 tune requires no less than 2,101,560 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 

 node of Neptune requires 1,958,692 years to traverse the circumference 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. 



We may here observe that the law which controls the motions and positions of 

 the perihelia of the orbits of Jupiter and Uranus is of the utmost importance in 

 relation to their mutual perturbations of Saturn's orbit. For, in the existing 



