SECTION I. 



VAKIATIONS OP THE ELEMENTS DEPENDING ON THE FHIST POWER OF THE 



DISTURBED FORCE. 



2. Let w, y, %, be the co-ordinates of Uranus, w, y', x', those of Neptune, x, y being 

 the plane of the ecliptic, m and m their masses, and let 



m (wa>'+ yy'+ *#') rri 



= (a?' 2 + y'*+ x'*)i~ ~ ^ _ x 'y + (y - y'y + (x-7f ' 



also let r = radius vector of Uranus, 



9 = longitude measured from mean equinox upon the ecliptic as far as the 

 node, and thence on the plane of the orbit, a, e, tit and e the semi-major axis, eccentricity, 

 longitude of perihelion, and epoch, n the mean motion in one mean solar day, i and SI 

 the inclination of the orbit to the ecliptic, and the longitude of the ascending node : and 

 let similar letters accented refer to Neptune. Then 



so = r {cos SI cos (9 - Si ) - sin SI sin (9 - Si ) cos i J , 

 y = r \ sin SI cos (9 - SI ) + cos SI sin (9 - SI ) cos i \ , 

 x = r sin (9 — Si) sin i, 



and similar equations hold for Neptune. 



Substituting these values of x, y, x, &c. in B, we have 



R = — {cos (0 - 9) + I\- , , 



where / = - 2 sin (9 - SI ) sin (ff - Si ) sin 2 1 i 

 - 2 sin (9'- Si ')sin (9 -Si ')sin 2 li' 

 + sin {9- Si ) sin (#' - Si ') sm * sin t' 

 + 4cos (a - a') sin (9 - a) sin {& -Si') sin s ^ i sin 2 -! i', 



and the equations for the variations of the elements of Uranus disturbed by Neptune, 

 are, putting n = mass of Sun + m, 



da 2nd 2 dR 



dt u. de 



de na y\ — e 2 

 dt fi£ 



(dR ,——dR\ 



\dis- de)' 



dur na s/l - e 2 dR 2na sin 2 1 i dR 



dt ixe de ^ s i n i y/i — e 2 di 



