1876.] 305 [Chase. 



VI. If contraction were to go still further, the centrifugal force of rota- 

 tion would give the nucleal particles orbits of increasing ellipticity. 

 These orbits would finally become infinite, when the average linear 

 motion was equivalent to parabolic perifocal motion ; or, in other 



words, when the mean linear velocity, Jl «2~| = [v^fr These ve- 



... Oj> n' ■^'^ 2 V ~fr 



locilies tend to or from coincidence at ~ — ; for X ~ X _ 



n'lz'' ^ - n 



= — ^V%fr = ^* 5^ ^^"fr' ^^ ^^"^ ^^^ *^^^* *^^^ limiting velocity between 

 complete diss:ciation and incipient aggregation (n K\/~fY), is - X the 

 limiting velocity between complete aggregation and incipient dissociation 



(n V fr ; see V.) 



VII. Let t = time of describing ^ r, in virtual approach to the centre, 

 under the action ot any central force/, = time of describing r in circular 

 revolution, or motion under constant pressure. Then/< = v^/r ; ?: ; = time 

 of free semi-circular oscillation; 7i ~ < = time of constrained semi-circular 

 oscillation = time of semi-rotation; nn ft = n iz y' fr = velocity ac- 

 quired in the time of semi -rotation = radial limiting velocity (V, VI). 

 Therefore the velocity of any central impulses which are capable of pro- 

 ducing aggregation, free revolution, and constrained rotation = velocity 

 l^roduced by constant equatorial pressure acting for one half-rotation. 



VIII. Radial oscillations, through a radius p = mr, give the central 

 dissociating velocity [ i/2 /r at -^^.- . 



IX. Undulations in elastic media tend to generate other undulations, 

 in arithmetic, geometric, harmonic and other figurate progressions. 



X. The limiting radius of free revolution (or the atmospheric radius 



a 

 in nebular condensation) varies as t^ ; the limiting radius of constrained 



1 

 rotation (or the nucleal radius) varies at t^. Therefore the nucleal radius 



OC 4 power of the atmospheric radius. In ordinary discussions of the nebu- 

 lar hypothesis, planetary aggregation has been regarded as atmospheric, 



under a velocity varying as \'— - ; but there are many traceable evi- 



' r 1 



dences of simultaneous nucleal activity, under a velocity varying as '^' 



ILLUSTRATIONS. 



Electrodynamics and thermodynamics furnish numerous illustrations of 

 Prop. I, but some of its most obvious exemplifications are found in cosmi- 

 cal revolution, atmospheric elasticity, axial rotation, and in the various 

 applications of Ferrel's laws. 



Proposition II identifies all central forces, so far as an identical ideal limit 

 of velocity is concerned. It is true that the ideal limit is physically unat- 

 tainable, but a full development of Peirce's theorj^ of vis viva and simul- 



