CERTAIN HARMONIES OP THE SOLAR SYSTEM. 75 



figure): "Let P be the point of equal attraction between Fig. 20. 



any planet and the next interior, the two being in con- ^^^ . \ 



junction ; P' that between the same and the one next ^ ^^ \% 



exterior. % Vl_ 



" Let also D = the sum of the distances of the points „ _^, ^/J__j\p1 



PP' from the orbit of the planet" (the whole PP' in the 'Si W "/ j^ 



figure); "which I shall call the diameter of the sphere ^/ /q l^ 



of the planet's attraction. ' ,,^ / '^ 



" D' = the diameter of any other planet's sphere of ' / 



attraction found in like manner. 



" n ^ the number of sidereal rotations performed by the former during one side- 

 real revolution round the sun. 



" 7i', the number performed by the latter ; then it will be found that 



(/) 9" 

 — ,y 



From this we shall have, alternately, 



n^ : D^ : : n'" : D'^ ; i.e. 



nr ^^'' 



r)3 ^^ "Tys ^ '^ consiant. 



The coincidence with fact is very close in the several instances of Venus, the 

 Earth, and Saturn. 



The proportion thus exhibited is analogotis to Kepler's 3d Law ; that the squares 

 of the periodic-times of the planets are as the cubes of their mean distances from 

 the sun ; and it is hence called KirhiooocVs A.nalogy. 



An " Examination" of this by the late Sears C. Walker is also given in the 

 Proceedings of the American Association for 18i9 (pp. 213 to 219 inclusive), and 

 its consistency with Laplace's Nebular Hypothesis made the subject of comment. 



Failure of the Analogy in the Case of Uranus. 



(107) Conceding that the time of rotation of Uranus [3 of (43)], as found by 

 W. Buffam, Esq., viz. 12 hours ±, is a first approximation to the truth; Kirk- 

 wood's Analogy will be found to fail in the case of Uranus. 



For if we apply Mr. Walker's formula, in which Q represents the time of rota- 

 tion (a mean solar day of the Earth being ===1); a, a planet's mean distance from 

 the sun; and D, the diameter of the (Kirkwood) sphere of the planet's attraction; 

 then, 



and we shall find, with the values of masses and distances as given in our Table 

 (A), in (3), that, in the instance of Uranus, 



=1'^.30380+ = 31.291 hours. 

 instead of nearly 12 hours; the result of the observation already quoted. 



