THE THEORY OF THE LONG INEQUALITY OF URANUS AND NEPTUNE. Hi 



d (nt + e) Zna* dR na y/l - e 2 , / r , dR 



., - n + - - — + - (y/l - e 2 - 1) — 



dt n da fxe de 



2na sin 2 A i dR 



H sin i y/i _ e i di ' 



na (dR . , . . (dR dR\] 



===== < -; — + 2 sin 8 i i [ — + - > , 



dt fi sin i v 

 d SI na dR 



dt fx sin i \/l - e* di 



3. In order to diminish the number of terms depending on the inclinations, we shall, 

 in calculating a, e, "sr, and nt + e, take the plane of the orbit of m at instant of epoch 

 as plane of reference, and then the first four equations become, putting u. = 1, and omit- 

 ting terms higher than the third order, 



, dR 



\nae -—. 

 * de 



- \nae (l - \ e 2 ) — ; 



and, putting i' = »y' and Si '= n', 



/ m - 2 sin (0' - n') sin (6 - v) sin 2 \ y 

 = {cos (d + ff - 211') - cos (9 - &)} sin 2 l 7 '. 



In calculating the inclination and longitude of node, we shall take the plane of the orbit 

 of m at epoch for plane of reference; therefore, putting y and II for the indination and 

 longitude of the ascending node on this plane, 



, IdR dR\ 



and / = {cos (9 + 9'- ZU) - cos (fl-ff)} sin 2 l 7 . 



4. In changing the plane of reference from the ecliptic to the orbit of m, it becomes 

 necessary to apply a small correction to the values of e' and ■&' as referred to the ecliptic, but 

 not to those of e and "zjt. Let PQR (fig. l) be the spherical triangle formed by the planes of 

 the orbits and the ecliptic, PR being in the plane of the orbit of m, QR in that of m, and 

 PQ the ecliptic. Now 9, e, and iff, are supposed to be measured from V upon the ecliptic to 



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