OF URANUS AND NEPTUNE. XXV 



28. The terms of the second order involving 2\, are by (Art. 8), 

 R, = m (P 2 cos 2X - .P/sin 2X), 

 where P 2 = iV,e 2 cos %■& + N 2 e' 2 cos 2-&' 



+ N 3 ee'cos (■& + tst') + JV 4 sin 2 1 y' cos 2 n', 

 P 2 = same with sines for cosines. 

 R 2 differs from R 2 only in having m for ml. 



Then the equations for the variations of the elements of Uranus, are 



. , (SrrirtfaP* m'na 2 dP 2 m'nae dP 2 \ . 



<>(nt + e) = r-7 jj + : — -, — ; — T , - sin 2\ 



\2to^sinl w sin 1 da 4a> sin 1 del 



+ (same with P 2 ' for P 2 )cos2\, 



l m'na dP 2 mnae 1 dP„' m'naeP 2 '\ . 



$e = ; — Tl —~ + : — n —~ ; — r, sin 2\ 



V 2a) sin 1 de 4o> sin 1 de 2w sin l / 



— (same with P 2 for P 2 ') cos 2A, 



. / m'na dP„ m'nae' dP 2 \ 



edW = ; jf -t— + ; 7j — — Sin 2\ 



\ 2<o sin 1 de 4w sm 1 de J 



+ (same with P 2 for P 2 ) cos 2A, 



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



$n = sm 2\ + - — cos 2X, 



to to 



. 2m'na 2 P 2 . 2m'na 2 P 2 



6a = sin 2\ cos 2A, 



a) to 



. / m'wtani-'y a dP 2 m'n tan A-y aP 2 ' m'nae tan 1<y dP 2 '\ . 

 *V = - ,— 3— + A-h, + ; ~— ~ Sin.2\ 



\(o sm l vi - e 8 sln 7 »7 w sm l 2«j sin l de / 



+ (same with P 2 for P 2 ) cos 2X, 



«„ m'n a dP 2 . 



d\\- ; a . — - sm 2A 



8oj sin 1 y/i _ e 2 sin* ^7 dU 



m n a dP 2 



cos 2\, 



8(o sin l" y/i -e 2 sin 2 l7 dU 



and those for Neptune are 



... fc / 3mri 2 a'P 2 mn'a' 2 dP 2 mn'a'e' dP*\ . 



6 (n't + e) = r-; — ^ + — r— 77 tt : — r, tt sm «X 



\ to sin 1 tosinl da 4wsinl de I 



+ (same with P 2 for P 2 ) cos 2A, 



. , ( mn'd dPo' mn'a'e' 2 dP 2 mn'a'e P 2 '\ . 



be = [- - — : — j. —r-r + : — r , — — r + : — — sin 2\ 



\ 2to sin 1 de 4w sm 1 de w sin 1 J 



— (same with P 2 for P 2 ') cos 2\, 



,« , / mn'd dP„ mn'a'e' 2 dP 2 \ . 

 e 6T7 - I - = — r-^T; -^ + - . .„ -r-r ) sin 2\ 



2to sin l" de 4<o sin l" de' / 



+ (same with P 2 for P 2 ) cos 2\, 



. , 6mn' 2 a'P 2 . 6mn' 2 a'P. 2 



Sn = sm 2A cos 2\, 



[D] 



