202 Prof. P. E. Chase on Harmonic 



equal intervals at the outer limit as well as two at the inner 

 limit, and the intermediate positions follow Bode's progression. 

 If we determine, by the method of least squares, the law of 

 planetary progression, we find that it is suggestive of har- 

 monic parabolical subsidence. 



Suppose Sun to be in the focus of a paraboloid of revolution, 



with a directrix-plane at tj— and a vertex at ^ — . Suppose 



3p bp rF 



the nearest fixed star (presumably a Centauri) to be in the 

 axis of the same paraboloid. Take 39 numerical abscissas of 

 the form A w =f^^ 2 , with A =?=■£■, A 19 =^L, A 38 = LM-r- 

 p = 7r 2 IA One third of the abscissas (A ..A 12 ) are within 

 the solar photosphere ; one third (A n . . A 25 ) are extra-solar 

 and inter-asteroidal ; one third (A 26 . . A 38 ) are extra- asteroidal 

 and inter-stellar. The next abscissa (A 39 ) is in a region of 

 predominating stellar influence, approximately, and perhaps 

 exactly, in the locus of a Centauri. The twenty-seven extra- 

 solar and inter-stellar abscissas may also be divided into three 

 equal suggestive groups, A 12 . . A 20 being inter-planetary ; 

 A 2 i . . A 29 having significant planetary relations ; A 30 . . A 38 

 being extra-planetary. The middle group (A 21 . . A 29 ) repre- 

 sent, respectively, J Mercury, ^ Venus, § Earth, J Mars, 

 f asteroid, f Jupiter, f Saturn, J Uranus, J- Neptune, the 

 indicated loci being all within orbital limits. We find here, 

 as in the Bodeian series, two equal numerators at the outer 

 limit, where the harmonic mean of the two coefficients is unity, 

 as well as two at the inner limit, where the harmonic mean is ^. 

 The coefficients -J, f , &c. represent successive and progressive 

 harmonic rupturing tendencies, inasmuch as particles falling 

 toward a cosmic focus from a distance nr would acquire the 



n't 



dissociative velocity v2gr at — — -= r. The reciprocal character 



of the Saturnian and Neptunian coefficients furnishes an indi- 

 cation of such retrograde tendencies as we may naturally look 

 for at the outer limits of a planetary system. 



Although it is impossible at present to anticipate with 

 certainty the precise way in which undiscovered harmonic 

 influences will be manifested, it may be possible to show the 

 probable existence of such influences and where to look for 

 them. The tendency to make absolute any close approxima- 

 tion to simple numerical relations, which is found in Jupiter's 

 satellites, should likewise prevail in planetary motions. The 

 number of such tendencies among the cosmical masses and 

 positions is so great that it is difficult, for want of definite 

 criteria, to judge of their relative importance. It may perhaps 



