Chase.] bo-i [Oct. 15 and Dec. 3, 



sideuce oi' mean approach towards the centre of preponderant attraction ; 

 while those which conspire, or which remain outstandiri<2^ after such con- 

 flicts, must ultimately give rise to circulation of a permanent character." 



In the earliest stages of nucleal aggregation, when the primitive 

 oscillating velocity subjects all particles to nearly equal impulses from 

 every direction, but with a slight preponderance towards special nucleal 

 centres, the variation from constancy of force may be so slight as to in- 

 troduce a geometrical progression based on the above thermal ratio, 

 1 : 1.4233. Since the nucleal radius of a Sun which would rotate syn- 

 chronously with planetary revolution varies as i/~J, while the planetary 

 radius-vector, or radius of possiile nebular atmosphere, varies as (ijS, the 

 atmospheric radius varies as (nucleal radius) s. "We have thus a basis for 

 the geometrical series, r, rj, n . . . r'"^. Now 1.42333 = 1.6009, or 

 almost precisely the fundamental radius (1.6007) which Professor Alex- 

 ander has pointed out in the arrangement of the Jovian system.* 



It is also very nearly represented in the ratios between the nucleal 

 radii of the inner pairs of planets, of the two principal planetary belts ; 

 9 : ^ = 1.598 \ \i \ % ■= 1.576.t The next term of the series is 

 1.42323 = 1.8008, which is remarkably coincident, both with Professor 

 Alexander's fundamental ratio:}: for the solar system (1 : 1.8), and with 

 the "centre of explosive oscillation," or the linear centre of oscillation 

 between a primitive centre of oscillation and a linear centre of gravity 

 (I and %). If the involution is carried to the fifth quadrangular pyra- 

 midal number, 1.4233^^ = 331390, which is within less than three per 

 cent, of the half-modulus of light at Sun's surface, measured in solar 

 radii. The pyramidal exponent, 35, is also within less than three per 

 cent, of the possible solar atmosphere measured in solar radii ; within 

 less than five per cent, of the half-modulus of light measured in Nep- 

 tunian vector-radii ; and within less than three per cent, of the nucleal 

 radius of a nebulous Sun which would rotate in a year of Uranus. 



If these accordances are dependent upon the mutual interactions of the 

 five principal masses in our system (0, 2/, Ij , §, \P), we may reasonably 

 look for still further accordances between the products of masses, which 

 enter as factors into expressions of joint gravitating action. We accord- 

 ingly find the following equation between the triangular powers of plane- 

 tary masses, designating mean perihelion,mean, and mean aphelion, by sub- 



.6 

 -3 



ftp 

 script figures 1, 3, 3, respectively : | — ^ 



1^3 



' ^^3 



X -3 



X 



* statement and Exposition of Certain Harmonies in the Solar System (Smithsonian 

 Contributions, 280), p. 15. 



The simple ratio, 1.4232 is approximated in the nucleal radii of the outer pairs ; 

 C^ ^ © = 1..373 : t|; -^ g = 1.4. 



I Op. cit., p. 4. I am informed by Prof. Alexan er that he announced this ratio be- 

 fore the American Association in 1857. 



