OF URANUS AND NEPTUNE. liil 



# (n't + e') = - o"032 sin 2\ + 9"-5S6 cos 2X, 

 + o"*020 sin 3X + l"-032 cos S\. 



The corresponding corrections of he, eh"sr, &c, are 



Se = - 3"-15 sin 2\ + o".65 cos 2X, 



e3V = - l""00 sin 2\ - 3" -08 cos 2X, 



3V- - 2"-33 sin 2X - 2"-52 cos 2\, 



e'$ w '= + 3"-86 sin 2\ - o"-91 cos 2X. 



The corrections for Aw, Aw' will be 



X (nt + e) = + 2"-017 sin 2\, 

 # (n't + e) = - l"- 44 sin 2X. 



The whole of the corrections of $n, In', for Ae, A^jr, Ae, A-ar, An, An, Aa, Aa, 

 will be 



$n m - o" -000591 7 sin X - 0"0001205 cos X 



- 0"-0001207 sin 2X - 0"-0000086 cos 2X 



- 0"-0000180 sin 3X. 



%n' = + 0"-0004360 sin X + 0"'0000889 cos X 

 + 0"-0000881 sin 2X + 0"-000006l cos 2X 

 + 0"-0000132 sin3X. 



59. Instead of the values of e and e given in Articles (5) and (6), we must use e + Ae and 

 e'+ Ae' in calculating the values of •£ and ¥ in the formulae of Art. (38), where 



Ae = + 2783"-82 sin X"- 42l"'24 cos X" 

 + 125 -02 sin 2X"- 3 -62 cos 2X" 



+ 8 -15 sin 3X"- -44 cos 3X" + (46 x 365 + 11) Aw 

 = + 2739"-88 - 3&' -05 = + 2703" -83 = + 45' 3" "83, 



and Ae'= - 1996".66 sin X' + 302"-14 cos X' 



- 89 -36 sin 2X' + 2 -60 cos 2X' 



- 5 -80 sin 3X' + '31 cos 3X' 



- - 1928"-81 = - 32' 8" -81. 



And instead of nt, n't, the corrected values (n + An) t, (w'+ Aw') t must be used. 



60. It remains now to correct the coefficients of the principal part of the long inequality 

 for the secular variations of the eccentricities and longitudes of the perihelia. 



Let E and H be the secular variations of e and "ar in one day, resulting from the action of 

 Jupiter, Saturn, and Neptune, and let E', H' be those of e and sr' due to the action of 

 Jupiter, Saturn, and tJranus. Then, substituting e + Et, w + Ht, e'+ E't, ■&■'+ H't for 

 e, w, e, sr', in terms of the first order in eccentricities, and integrating, neglecting squares 

 and products of EH, E'H', we have, 



