1876.1 ■^^^ [Chase. 



21. The laws of central forces require that provision should be made for 

 radial oscillations, tending towards the time-limit of iso-radial circular 



oscillations (|/32and j/S); for tangential velocities, varying inversely as 

 the times and directly as the fourth root of central iso-radial tendencies; 

 for centres of oscillation in lines of force, and for oscillations between sys- 

 tematic and locally dominant centres. We have already seen (15, 17, 18,) 

 how closely the relative masses of Sun and Jupiter provide for the last 

 requirement; if they provide also for the others, the centre of oscillation 



for Sun's possible atmosphere should be at 1047.875 — (/Ss)* = 23.875 

 solar radii . The corresponding height of possible atmosphere, or the height 

 of equality between the velocities of rotation and revolution, is f of 23.875 

 = 35.818 r. This would give, for the time of solar rotation, 365.256 -r- 



(214.86 -^ 35.813)'' = 24^.856, difiering less than ^ of one per cent, from 

 the vis vica estimate (20). 



22. The ratio 1 : 32^ is also simply connected with the mass and distance 

 of Jupiter's companion planet, Saturn, and so with the centre of planetary 

 inertia. For 1024 + 1025 + .536 = 2049.54 (3,17), Saturn's mean radius 

 vector being 2049.51 solar radii; 1024 + (35.813 •-- /.4) = 1080.625 is the 

 limit, of which Jupiter represents a centre of explosive oscillation, and the 

 inertial moment at the limit gives the mass of Sun -;- Saturn; for 1080.- 

 625 X (5)'^ = 3501.2, Bessel's value being 3501.6. 



23. Among the many harmonies of planetary mass which manifest a 

 dependence on nebular influences, the following are, perhaps, indicative of 

 some of the earliest forms of activity. 



a. The masses of Jupiter and Earth are nearly proportionate to the 

 squares of their periodic times X the velocities due to internucleal vis viva; 



5.2028'' X 5.2028^ = 321.2; 321.2 X 1047.875 = 336201. 



b. The influence of spJierical gj'-ration on Venus (11), seems to be further 

 shown by its ratio to Earth, which is the square of the ratio of Uranus to 



Neptune (|/8 : ~) : "^ X 336201 -- 8 = 415289. Hill's estimate of the 

 mass of Sun -=- Venus is 408134. If the internucleal vis viva of Jupiter 

 were taken at secular perihelion, the resulting theoretical mass-denomi- 

 nators would be Earth, 326222; Venus 402460. 



24. The masses of the principal planets, therefore, seem to have been 

 primitively determined by the following influences: — Neptune, by the 

 proportion between the time of direct fall to the centre of planetary inertia, 

 and the time of circular revolution; Uranus by the time of describing the 

 same proportional part of a circle, in the circular orbit; Saturn, by equality 

 of nebular vis viva with Jupiter, when the two centres of condensation were 

 in opposite parts of the nebular belt and on opposite sides of the sun; 

 Jupiter, by the ratio of variability, between incipient fall to a centre of 

 linear atmospheric oscillation and circular revolution ; Earth, by the 

 combined action of vis viva and time of revolution; Venus, by the action. 



