100 Mr. Lubbock on tlie Stability of the Solar System. 



and indeed however far the approximation be carried. So that 

 the following theorem is true, without neglecting any powers 

 of the eccentricities or inclinations, or any powers of the dis- 

 turbing forces : 



Semi-major axis ~) 



Eccentricity o c - - \ 



Inclination of the orbit to a >= Senes of ">. smes th0 " 



fixed plane j ".'^ 'I 1 ' 3 " 11 '? 



by the time. 



Longitude of the perihelion ~) 



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



Longitude of the node ) tity multiplied by the time. 

 The series of cosines being a periodic function, it follows that 

 however long the periods of some of the inequalities may be, 

 the semi-major axis, the eccentricity, and the inclination to a 

 fixed plane vary periodically within limits which depend upon 

 the magnitude of the disturbing forces, that is, upon the mag- 

 nitude of the mass of the primary compared with the masses 

 of the planets, and upon their mean distances from the primary, 

 &c. The other three constants have a term varying with the 

 time ; but this, from the nature of these constants, does not 

 affect the stability of the system. 



The contrary obtains when the body moves in a medium 

 which resists according to any power of the velocity, in this 

 case, considering only the terms which depend on the resistance 

 of the medium ; 



Semi-major axis 1 . - . 



Eccentricity ) = Sene , S <**f + a , qUant ' ty 



Longitude of the perihelion \ Q mult 'Plfd by the time. 



Lonfritude of the epoch / : = Senes of . cosm f s . *^ 



<* any quantity multiplied by 



the time. 



I have also extended these results to the problem of the ro- 

 tation of the earth about its centre of gravity. The solution of 

 this problem contains six constants: which constants are analo- 

 gous to those which occur in the determination of a planet's 

 motion round the sun, an analogy first, I believe, pointed out 

 by M. Poisson. 



* By integrating the expressions for the variations of these 

 constants, 



The mean motion of rotation, ~] 

 The cosine of the geographical j 



latitude of the axis of install- )>= A periodic function with- 

 taneous rotation, out any quantity multi- 



The obliquity of the ecliptic, J plied by the time. 



The 



