Ghase.J '^Oo [Jan. 7, 



by a body in falling from a distance 7ir to any other distance x may be 

 found by the equation, vdv = g)'-x~^d {nr — x) ; integrating and reducing, 



( 2 gr {nr — .r) ) 2 



V ^= < >- • Ifthis velocity IS due to a srnchronism of activity, 



( nx j 



between nebular condensation within any given stellar system and parabolic 



nr — X nr 



projection between dfterent systems, «=: /S^'?' ; ^^ — 1 ; .r — ^ ^ 



n 2 



77. If V =z 2, ==-sr. This corresponds with the centre of linear os- 



' ,1 4- 1 3 



cillation ; it is also the locus of belt formation, on account of the collision 



of subsiding particles from opposite extremities of a nebular diameter. In 



such subsidence there would be a tendency to form confocal elliptic orbits,* 



Br — 



with major axes of — and minor axes of r y^2. 



0. In rapturing or explosive oscillation, the vis viva of wave propaga- 

 tion is f of the vis viva of oscillating particles. f This relation was subse- 

 quently pointed out by Maxwell, :|; apparently witliout being aware that I 

 had called attention to it five years previouslj^ 



1. Harmonic nodes with the ratio ^, (5), indicate tlie primary division of 

 the solar system, as may be seen by tlie following comparisons, in which r^, 

 represents Sun's radius : 



Theoretical. Observed. 



927-0= 81 7-0 Mercury 83.05/-,, 



937-0= 729 ?•„ Asteroid 76 731.37 /•„ 



9* ?'o = 6551 ?-o Neptune, s. a.§ 6536.91 r^ 



IL The influence of subsidence (-/y), at the outer surfaces of the two 

 primitive belts, is shown as follows : 



Theoretical. i~)b.served. 



I Neptune = 20.0226 Uranus, m. a., 20.04*2 



I Mars = 1.0158 Earth, m. a., 1.0339 



We have already seen, (3), that the middle of the bell of greatest con- 

 densation is 1.0169. 



III. The influence of rupturing oscillation, {0), at the inner surfaces of 

 the two primitive belts, is equally evident. 



Theoretical. Observed. 



I Mercury = .696778 Venus, m. p., .696779 



I Jupiter = 9.3650 Saturn 9.5389 



The oscillating limits of all portions of the .Jupiter belt are within the Sa- 

 turnian belt. 



*Proc. Am. Phil. Soc, xvii, 98-109. 



tib. xii,392-J. 



X See Phil. Mag., .June, 1877, p. 4.jo. 



? s., secular; a., njohelion ; m., mpan ; \).,perhelio)i. 



