THE THEORY 



OF THE 



LONG INEQUALITY OF URANUS AND NEPTUNE. 



1. If the mean motions n and ri of two planets be nearly in the ratio of two 

 small integers k' and k, then kn^-k'n is small compared with n and n', and therefore 

 those terms of the disturbing function which involve (kn — k'n') t in their arguments will 

 be much increased by integration; and since k-^k' is small, the principal terms of that 

 form are of a low order. The variations of the elements, therefore, depending on such 

 terms, are of considerable magnitude, and occupy a long period of time in going through 

 all their changes. 



Now the mean motion of Uranus is nearly equal to twice that of Neptune. Hence 

 there is a long inequality in the elements of these planets, the principal part of which depends 

 upon terms of the first order in the eccentricities involving (n - Zn) t in their argu- 

 ments. There are also terms of the third order in eccentricities and inclinations depend- 

 ing on the same angle, and terms of the second and third orders involving 2 (n - 2n) t 

 and 3 (n - 2ra') t respectively, all of which are sufficiently important to be retained. The 

 object of the following essay is to calculate the variations of the elements depending on 

 these terms ; then, to calculate the principal terms depending on the square of the dis- 

 turbing force ; and, finally, to give some explanation of the mode in which the dis- 

 turbing forces act in producing the long inequality. 



[A] 



