CEETAIN HARMONIES OE THE SOLAR SYSTEM. 



SECTION I. 

 INTEODUCTOET. 



(1) Keplee's 3d Law is ordinarily expressed by saying that the squares of the 

 periodic times of the several planets of the solar system are to one another, respec- 

 tively, as the cubes of their distances from the sun. The same law includes also 

 the periodic comets, and it is, in like manner, applicable to the satellite systems. 



But from this we do not learn that any laws are to be found determining the 

 ratios of the distances themselves. 



It will be one main object of the present discussion to show that such laws exist, 

 and precisely what they are — generality and precision being characteristics of every 

 law of nature.^ 



(2) Approximations to the laws in question have, from time to time, been ex- 

 hibited, by the author of this paper, to the American Association for the Advance- 

 ment of Science, at several of their meetings, beginning with that at New Haven, 

 in 1850, and more especially, also, that at Montreal, in 1857; Baltimore, in 1858; 

 and Springfield, Massachusetts, in 1859 ; but it is only within the past few months, 

 or even almost up to this present time (July, 1874), that the entire form and 

 consistency of the results hereinafter exhibited have been quite fully made out. 



(3) All that is to be stated will, it is conceived, be the more readily intelligible 

 by proceeding, as occasion may seem to require, inductively, and consequently fol- 

 lowing, to some extent, the order of discovery. 



Antecedently even to this, however, it seemed to be desirable to discuss anew 

 the expressed values of the distances in question, and this, in view of the fact, 

 that Kepler's 3d Law is itself slightly modified by the consideration due to the 

 masses of the revolving bodies. 



Thus if ilf represent the mass of the sun, and m,in' i\\e respective masses of any 

 two planets, while a, a! represent their mean distances from the sun, and T, T' 

 represent their periodic times, we have 



\f)=\a) ^ffqr^"°^- I 



/TV if + m ' _ /ay r • • • ( ) ' 



Kt) ^ W^m ~\al J 



* The so-called Law of Bode or of Titius, it need scarcely be said, fails in both these respects. 

 1 November, 1874. f 1 ) 



