1875.] 



653 



[Chase. 



or, if we introduce all of the five great masses: 



There is still so much uncertainty as to the masses of Neptune and 

 Uranus, that it is impossible to tell how close this agreement may be, 

 but the deviation from precise accuracy cannot be large. According to 

 Newcomb's latest determinations of those masses, the equation gives twO' 

 values for Saturn, one of which is slightly 1 irger, the other slightly 

 smaller, than Bessel's value. By looking a little further we may find 

 relations which cin be measured with greater certainty, and are there- 

 fore moi-e satisfactory. 



La Place found that if the mass of each planet be multiplied by the- 

 pi'oduct of the square of the eccentricity and the square root of the mean 

 distance, the sum of all the products will always retain the same magni- 

 tude ; also, that if each of the masses be multiplied by the prbduct of 

 the square of the orbital inclination and the square root of the mean dis- 

 tance, the sum of the products will always remain invariable. Now the 

 square root of the mean distance varies inversely as the velocity of cir- 

 cular revolution at the mean distance, or inversely as the square root of 

 the velocity of nucleal rotation at the same distance. It is therefore 

 probable that the primitive undulations may have influenced the relative 

 positions as well as the relative masses of the principal planetary orbs. 

 Stockwell has found* the following relations : 



I. The mean motion of Jupiter'' s periTielion is exactly equal to the 7nean 

 motion of the perihelion of Uranus, and the mean longitudes of these peri- 

 helia differ by exactly ISQO. II. The mean motion of Juintefs node on the 

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

 of these nodes differ by exactly 180°. 



I have already had frequent occasion to refer to the position of the 



nebular centre of planetary inertia (-i/ v^,.2_^ v-^^^j 



in Saturn's orbit. 



, '\l Imr 



If the four great planets were ranged in aline, Jupiter on one side of the 

 Sun and the other planets on the other, tlie tidal influences, when Satu:n 

 was in mean position, would drive Jupiter, Uranus, and Neptune to, or 

 towards, their respective aphelia. Those positions would accord with 

 Stockw^ell's two theorems, they would approximate the centre of inertia 

 very closely to Saturn's mean radius vector, and they would make the 

 equation of the products of triangular powers applicable to vector radii, 

 as well as to masses. For the logarithms of mean vector-radii of the 

 four outer planets, according to Stockwell, f are : — 



Neptune, mean Aphelion, 1.481951 Neptune, 1.481951 



Uranus, " 1.301989 (Uranus)^^ 3.905957 



Jupiter, " .734588 (Jupiter)"^ 4.40752& 



10)9.795441? 



Saturn, mean, .979496 Saturn, .979545 



* Memoir on the Secular Variations of the Elements of the Orbits of the Eight Prin- 

 cipal Planets (Smithsonian Contributions, 232), p. xiv 

 t Ibid. pp. 5, 38. 



