OF URANUS AND NEPTUNE. XV11 



lo g ,"•",» ■ 4,4063 °77, log /"." ., = 3-7735706, 



a/ sin 1 or sin 1 



m'n „ . mn' 



log : Tl = 2*7235186 -, log : T , = 2-3828882 -, 



co sin 1 to sin 1 



log (a J - 2-41008, log (a + — J = T-97549, 



16. We shall now proceed to calculate the variations of the elements depending on terms 

 of the first order. By (Art. 8), 



P, = m'(P cos X - Psin X), 

 where P = Mi e cos w + ilf.e'cos -ar', 



P' = same function with sines for cos ; 

 and Ri — in (Q cos X - Q'sin X), 

 where Q = M(e cos -ar + M 2 'e cos w ', 

 Q' = same with sines for cos. 



When the equations of (Art. 3) are integrated, we have for the variations of the elements 

 of Uranus, putting co = n - 2ri, 



. /Sm'n'oP 2m'na 2 dP m'nae dP m'nae 3 dP\ . 



6 (nt + e) = -t— : — 77 + — ; — zr -; ; — w ~r + ■ 77 T - sin *• 



\to sin 1 co sin 1 da 2co sin 1 de 8co sm 1 de I 



+ (same with P instead of P) cos X, 



& 



(m'na dP' m'nae 2 dP m'naeP'\ . 

 : — 77 —r- + : — W —, : — W Sln X 

 w sin 1 de 2co sin 1 de 2co sin 1 / 



- (same with P for P') cos X, 



« I m'na dP m'nae 2 dP\ . 



eb-ur = - . ;, — + ~ r— 77 "J- sinX 



\ co sm 1 de 2m sin 1 de I 



+ (same with P for P) cos X, 



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



en — sin X H cos X, 



CO w 



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



oa = sm X cos A, 



to w 



. . m'nae tan i«y dP' m'no tan l«y . . , 



*y ■ ( : — „ —i : — » P) sin *• 



to sm 1 de w sin 1 



- (same with P for P) cos X, 



Sn = o ; 



and the variations of the elements of Neptune are 



, / 6mn' 2 a'Q 2mn'a' 2 dQ mn'a'e dQ mn'a'e' 3 dQ \ . 



+ (same with Q' for Q) cos X, 



, , / mn'a dQ' mn'a'e' 2 dQ' mn'a'e Q\ . „ 



5e = ( : — -. — r + : — T , -ry + . „ sin X 



V to sin 1 de 2 co sin 1 de co sm 1 J 



- (same with Q for Q') cos X, 



