202 I'l'oC. P. E. Cba.sc on Momentum and Vis viva. 



belt, wliicli ap})ears to have been thus broken up by the action of 

 UranuSj Saturn^ and Jupiter. By estimatin^r the condensation 

 from Neptune's mean distance, and from secular limits of Ura- 

 nuS; Saturn^and Ju])iter, we find the following accordances: — 



Neptune ... -^n = 2'b77 Astrsea =2"577 



Uranus s.p.*-f-n= 1-517 Mars =1-524 



hjaturn s. p. -Hn= -749 Venus a.*... = '749 



Ju})itcr s. a. -i-n= '473 Mercury s. a. = '477 



This leaves the orbital velocities of the four outer ])lanets to 

 be accounted for by like condensation from an earlier nebu- 

 lous condition of which we have no present visible evidence ; 

 but if the main hypothesis is correct, we may reasonably look 

 for confirmation of a different kind within the present limits 

 of the solar system. If we consider the vis viva of orbital and 

 of radial velocity for unit of mass, the vis viva added by radial 



fall from r to - is {m — V)gr, while the vis viva added by equi- 

 valent orbital contraction is only ^ (in — l)^r, or one half of 

 the radial addition. A simple nebular condensation from r to 



- would therefore add gr to the vis viva, which is equivalent 



. . . r . 



to the vis viva of circular-orbital revolution at r. There is 



therefore a tendency to repeated nebular rupture at ^? ^^ 05 ••• o^* 



Starting from the present outer limit of our system, Nep- 

 tune's secular aphelion, these rupturing-nodes would occur at 

 15-235, 7-017, 3-809, 1-904, -952, -476, -238 ! The first belt 

 would include Neptune and Uranus ; the second, Saturn ; the 

 third, Jupiter ; the fourth, the Asteroids; the fifth. Mars and 

 Earth; the sixth, Venus (grazing also the Earth and Mercury 

 belts) ; the seventh. Mercury. 



After the nebula had assumed a globular form, these rup- 

 turing nodes would occasion constant tendencies, from oppo- 

 site extremities of every diameter, to the formation of confocal 



Q 



elliptic orbits, with major axes of -7- and minor axes of ^/8r. 



2r 



Those ellipses would mutually intersect at -77-, thus tending, 



through collision of particles, to form a belt at that distance 



from the centre. The vis viva communicated by simple fall 



2r 

 from r to -^=\gr, which is equivalent to vis viva of circular- 



* a., mean aplielion ; s. a., secular aplielion ; s. p., secular periheliou. 



