xlvi THE THEORY OP THE LONG INEQUALITY 



,w , 8m,»m' 2 



Hence log a C = 0-37140, also log — — — T . = 1-13239, 



w sin 1 



.-. 57 = + i"-898 sin X - o"-274 cos A. 



For the action of Jupiter on Neptune, we have 



„ , , , s rd C8JT) 8m 2 mn' 2 a'W °* -, ^ 



57 - Zria* I — Vt-^ = -T-- — tt- (Q sm X + Q cos X), 



J t da to 2 sin 1 



where aD =b + 2a -p, + |a 2 — - %, 



and log a" = T'23859, log 6 " = 0-30433, 



, ,,db'' - . ,, o d?b " - , 8mW* 



log a" -4r = 2-49205, log a"" —^ = 2-52123, log - 1-65634. 



da da 2 a> 2 sin 1 



Hence log a'D' = 0-32098, 



.-. 57 » + 5"-647 sin X - 0"-815 cos X. 



53. The effect of the square of the disturbing force on the eccentricity and longitude of 

 perihelion, will be given by the equations 



na rl d($R) _ 5e dR\ _ r($e dR _ d gjffl 



e J t \ dip e dw) ' J t \ e de de 



(see Pontecoulant, Liv. VI. p. 213), omitting terms of the first and higher orders, except those 

 terms of the first order which will be divided by <o 3 . First let 



na r d (Jfi) SmVo 2 4m ax / dP" „ dP\ 



then — [-±-±m+ (!+ — «') [P -J--P ~\nt 



e J, dw 2<«r m \ de de I 



( I dP ,dP\ . \ 



3mW 4m J \ de del I 



"4a,3 s ini" (1+ ^ a) ) , d P w dP\ (' 



therefore, omitting the secular perturbation, 



5^ = - l"-267 sin 2X - 3" -413 cos 2X. 



« r d($R) SmWa? , 4m - /' dP „,dP\ 



,., ( (pf- *fW) 



Sm^a 2 4m 9X > V de de / f 



( + ( P ^ +iy d7) cos2X ) 



4o) sin 1 m 



.-. e5V p + 3" -413 sin 2X - l"-267 cos2X. 



