CERTAIN HARMONIES OF THE SOLAR SYSTEM. 47 



of the fourth (Dione) double that of the second (Enceladus). The coincidence is 

 exact in either case to about the 800th part of the larger period."^ 



Again, in the American Journal of Science and Arts, Sd Series, vol. iii, p. 67 

 (1872), is an extract from a letter of Prof. Benjamin Peirce to Prof. Newton, in 

 which Prof. Peirce says : " I have discovered three fixed equations between the 

 mean motions of the four outer planets. If the mean motions of Jupiter, Saturn, 

 Uranus, and Neptune are respectively represented by n"-', n''\ n'''\ and ?i'''", these 

 equations are — 



4?^" + 9ra"" = 16J^"' 

 2rt^ + 17?i"' + 6?i'''"=12n" 



3n"' + 8«"" = n" 



To which he adds " If all the three equations are admitted, 



the mean motions of three of these planets can be computed when the fourth is 

 given;" and he exhibits the requisite equations. He states, moreover, that the 

 reception of these " involves a laborious revision of the theory of these planets, 

 and must seriously change the elements of their orbits." 



Lastly ; — to this. Prof. Daniel Kirkwood adds :^ " The recent note of Prof. Peirce 

 announcing his discovery of some interesting relations between the mean motions 

 of the four outer planets, has recalled my attention to a number of similar coinci- 

 dences detected by myself several years since, while engaged in a somewhat labori- 

 ous examination of the planetary elements. Of these the following may be worth 

 putting on record for future discussion : — 



2ra^ _3 K" _ lin"" =0 (1). 



2/i"— 21?i^" + 30?z"" =0 (2). 



3n^ — 8 n"' - 2«^" + 7w"" = (3). 



" The re-examination of the last of these has recently led to the discovery of two 

 others, viz : — 



68w" - 325w"' + 257?i™' = (4). 



257?i^ — 844«" + 587«''" =0 (5)." 



" The fifth, however, is not an independent equation, but is derived 



from the third and fourth. ... It is obvious, moreover, from the same equations, 

 that no three of the four outer planets can ever he in conjunction at the same time." 



The more thorough revision indicated by Prof Peirce would be requisite before 

 all these relations could be definitely settled ; but they furnish additional occasion 

 both in the planetary system and in that of Saturn for the explanation which M. 

 Laplace himself has given, in Note VII to the Systems du Monde, of the special 

 relation apparent in the first of the instances here quoted, viz., that of Jupiter's 

 satellites. 



That illustrious astronomer indicates that " in order to produce the equation 

 with regard to those satellites, already quoted, it would be sufficient that, at first, 



^ Outlines of Astronomy (lltli edition), (550). 

 " At p. 208 of the same volume. 



