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XVIII. On the Stability of the Solar System. By J. W. LUB- 

 BOCK, Esq. F.R.S.* 



THHE following passage occurs in the 103rd Number of the 

 * Edinburgh Review, p. 4-3, lately published. " The earth 

 is one of eleven planets which revolve round the sun. It has 

 been demonstrated by mathematicians, that all the little irre- 

 gularities arising from the mutual actions of the planets on each 

 other run through regular periods, and then vanish. So that 

 their motions, for anything which we know to the contrary, 

 may continue for ever, without any real alterations in the mu- 

 tual distances between the sun and planets." 



The proof of this proposition, as here stated in its utmost 

 generality, is not to be found in any work on physical astro- 

 nomy ; nor is it true, unless the planets move in a medium ab- 

 solutely devoid of any resistance. The proof given by M. de 

 Pontecoulant, Theorie Anal, du Sy steme du Monde, vol. i. p. 4-55, 

 extends only to the square of the disturbing force. In rigour, 

 however, it matters not at what stage of the approximation 

 the terms come in which create a derangement; the effect might 

 be more slow, but would not be less certain. 



In a paptr recently published in the Philosophical Trans- 

 actions, I have endeavoured to overcome this difficulty by the 

 following very simple considerations. 



By th3 first approximation, or that which takes into account 

 the first power of the disturbing force, supposing the body to 

 move in a medium devoid of resistance, 



Semi-major axis "1 



Eccentricity I = & , f cosines . h 



Inclination of the orbit to a f . w ]fed by 



fixed plane J tfjfos/ 



Longitude of the perihelion^ 



Longitude of the epoch >= Series of sines -f a quan- 



Longitude of the node J tity multiplied by the time. 



The arguments under the sign sine and cosine in these ex- 

 pressions are multiples of angles depending on the mean mo- 

 tions of the bodies which compose the system. 



A second approximation may be obtained by integrating 

 the differential equations for the variations of the elliptic con- 

 stants, after having substituted in the disturbing function their 

 values found by the first approximation. But the values thus 

 found for them by the second approximation retain the same 

 form as before : the same is true for the next approximation ; 



* Communicated by the Author. 



O2 and 



