1873.] li/7 [Chase. 



17. Centres of linear oscillation are found within the asteroidal belt 

 at the oppositions of Neptune and Uranus, Neptune and Saturn, Uranus 

 and Jupiter. The immense inertia .of the Sun seems, in each of these 

 cases, to have disturbed, without wholly destroying, the tendencies to 

 planetary aggregation. 



18. A careful examination shows numerous other perihelion, mean, or 

 aphelion distances, connected with centres of linear oscillation, some of 

 which I propose to enumerate in a subsequent communication. 



10. Saturn's mean aphelion (10) is a mean proportional between 

 Jupiter's mean perihelion (4.978) and the mean aphelion of Uranus 

 (20.043), as well as between Uranus's mean perihelion (18.322) and 

 Jupiter's mean aphelion (5.427). 



20. The geometric mean of the theoretical harmonic planetary distances 

 (2.13716), varies less than one-fortieth of one per cent, from the geometric 

 mean of the actual planetary distances (2.13665). This approximation 

 corresponds in precision to the one given, in my paper on "The Gamuts 

 of Sound and Light," between the mean wave length of the Fraunhofer 

 lines and those of the musical Scale. 



21. The inner limit of the asteroidal belt (Flora = 2.2) is nearly coin- 

 cident with the geometric planetary mean. , 



22. The year at the harmonic theoretical aphelion of Mars (1.659" = 

 2.1376), is to the terrestrial year, as the geometric planetary mean is to 

 the Earth's mean radius vector. By means of this ratio, Pierce's Phyllo- 

 tactic series may, perhaps, be connected with the harmonic series. 



23. The linear centre of oscillation of Jupiter's radius vector, and the 

 shorter extremity of the solar-jovian linear pendulum of which the Sun 

 occupies the centre of oscillation, are both in the asteroidal belt. 



24. The potential of vis viva in a gravitating mass (mv l ) varies as the 

 product of the mass by the distance (md, or at the surface, mr), there- 

 fore the product of the mass by the radius may, perhaps, represent the 

 limiting excursion of communicated elastic undulation at the surface. 

 Such is at least the case, if the theoretical velocities of sound and light 

 are the limiting velocities at the Earth and Sun. 



25. The velocity communicated, during a given angular rotation, by 

 Earth and Jupiter (the controlling orbs of the intra and extra-asteroidal 

 systems) to falling bodies at their surface, is to the velocity communi- 

 cated by the Sun, in an equivalent angular rotation, as planetary velocity 

 at the Sun's surface, is to the velocity of light. 



26. The centres of linear oscillation between Neptune's mean aphelion 

 and Uranus's mean perihelion (2.1019), and between Neptune's perihelion 

 and Uranus's aphelion (3.4503), are near the apparent inner and outer 

 limits of the asteroidal belt. 



It gives me great pleasure to acknowledge my indebtedness to Stock- 

 well's discussion of the secular variations of the planetary elements, for 



