XXV111 THE THEORY OF THE LONG INEQUALITY 



m'n tan 1 y a dP 2 ' 



\/i - e*w sin l" sin 7 dy 



= - 1-4979, 



and, omitting the multiplier , which = l very nearly, 



VI - e' 2 



mri a dP 2 „ „ mri a dPj ., 



: » T-n — > TFF> = + 40'-6591, : w 7-7= — , — 4 = + 35-5233, 



8wsinl" sin^y dll 80, sin l" sin 2 ^ dll 



»rara' tan A »y' a ' dP 2 „ nrn' tan A <y a' dP 2 ' ,, 



H^ " ' T - ' = + ° 9343 ' H^ 1 2 ' *7 " " x 069 *' 



a, sin 1 sin y dy w sin 1 sin 7 07 



m'ra tan A 7d? 2 ,, »i'» tan 1 yaP,' ,, 



r-^7 = + 0"'0242, tMt - - 0"-0008, 



w sm l oj sin 1 



m'nae tan JL -y dP» „ m'nae tan 1 -v dP 2 ' 



; «_£ -— = + O'-0200, ; ^-i — -^ = - 0"-0043, 



2w sm 1 de 2o) sin 1 de 



2mn tan 1 y a P 2 „ „ 2mn tan^aPj „ 



;-H-£ = + 0-0346, r*-5 = - 0-0011, 



cu sin 1 w sm 1 



mria'e tan 1 <y' dP, „ mn'de tan 1 <v' dPl 



: — W- 1 - -rr = + 0-0027, ; — £-= rr = + -0029- 



2a, sm l" de 2w sin l" de 



Therefore, 



&y = + i"-503 sin 2\ + l"-353 cos 2\, 



^n = - 49"-760 sin 2X + 56""954 cos 2X, 

 $y'= + i"-065 sin 2\ + 0"-902 cos 2\, 

 5n'= - 35" -523 sin 2\ + 40"-659 cos 2X. 



33. For terms of the third order involving 3\, 

 .R 3 = m (P 3 cos 3\ - P 3 sin 3X), 

 where P 3 = K^e 3 cos 3tjt + K 2 e' 3 cos 3nr' + K^re cos (2sr + ■& ), 

 + JT 4 ee' 2 cos (27«r'+ 73-) + i^e sin 8 ^ y cos (2n'+ -sr) 

 + .ST/ sin 2 1 7' cos (2ll'+ -ar'), 

 P 3 ' = same with sines for cosines. 



R 3 differs from R 3 only in having m for tw'. 



The equations for Uranus are 



» , fm'n s aP 3 2m' na? dP 3 m'nae dP,\ . 



6 (nt + e) = -r—. jj + : Tt — : Tl —r- Sin 3\, 



v ' \u3' sm 1 3a, sm 1 da 6 a » sin 1 de / 



+ (same with P 3 ' for P 3 ) cos 3X, 



* w'«« d/Y . m'na dP 3 



Ce = r— 77 — — sin 3\ + ; -— COS 3X, 



3w sin 1 de 3ft, sin 1 de 



. m'«a dP 3 . m'na dP 3 



edsr = ; — jr — — sin 3\ : — r . - -~ cos 3A, 



3ft, sin 1 de 3ft, sin 1 de 



. 3m'n 2 aP 3 . 3m'n 2 aP 3 



bn = sin 3\ + cos 3X, 



