XXX11 THE THEORY OP THE LONG INEQUALITY 



$ri = + o"-00129446 sin (2* - $ ) - o" -00900457 cos (2^ - # ) 

 + 0"-0O002737 sin 2 (2S> - W) - o"-00082441 cos 2 (2* - $ ) 

 + 0"-00000503 sin 3(2^ - $) - o"-00008279 cos 3 (2^ - $). 



$d = - '00120254 sin (2* - W) + 00836513 cos (2¥ - #) 



- -00002543 sin 2(2^ - #) + -00076587 cos 2 (2^ - W) 



- -00000467 sin 3 (2^ - ¥) + -00007691 cos 3 (2^ - #). 



&y' = _ 0"-285 sin (2^ - $ ) + 0"-083 cos (2* - $) 



- l"-065 sin 2 (2* - # ) + o"-902 cos 2 (2* - $) 



- 0"-155 sin 3 (2*- #) + 0"-139cos3(2^ - # ). 



^11' = - 55"-703 sin (2* - $) + 8" -242 cos (2* - ¥ ) 

 + 35"-523 sin 2 (2^ - $) + 40 "-659 cos 2 (2* - #) 

 + 5"-299 sin 3(2* - W) + 5"-883 COS 3(2* - #). 



The period of the principal part of the long inequality = — ; x periodic time of 



2ra — n 



Uranus, or about 4047 years ; and the periods of the smaller terms depending on 2\ and 3\ are 



respectively one half and one third of the period of the principal terms. 



SECTION II. 



VARIATIONS OP THE ELEMENTS DEPENDING ON THE SQUARE OP THE 



DISTURBING FORCE. 



39. In order to obtain the terms depending on the square of the disturbing force we must 

 substitute the first approximate values of the elements, a + $a, e + Se, &c. instead of their con- 

 stant values a, e, &c. in the differential equations for the variations, and then integrate afresh 

 rejecting the terms of one dimension in the masses. 



First, let us consider the effect on the principal part of the long inequality in mean lon- 

 gitude. 



Let £ = this part of the inequality, then 



Then substituting n + $n, a + <$a, &c. 



n 2 a = -5 , .-. § (ra 8 a) = - 2n?8a, 

 a 



s a 2 r dR 



da = - 2nar / — - , 

 J i de 



.-. 8 (w 8 a) = 4»W f — , 

 J. de 



