OP URANUS AND NEPTUNE. lv 



X (n't + e') = (- 49"'51 + 0"-007 sin (2* - $ ) + (l07"-87 + <>"'032 *) cos (2* - # ) 

 + 8"-37 sin 2 (2* - # ) + 12 "-S3 COS 2 (2* - # ) 

 + l"-66 sin 3 (2* - # ) + l"-03 cos 3 (2* - $ ). 



# e = + i7"-45 sin (2* - $ ) + 8"-08 cos (2* - # ) 

 + 6"-10 sin 2 (2* - # ) - 7"'57 cos 2 (2* - # ) 



eSV = + ll"'62 sin (2* - # ) - 14"-52 cos (2*-#) 



- 10"-72 sin 2 (2"^ - $ ) - 7"-71 cos 2 (2* - $ ) 



- l"-32 sin 3 (2^-#). 



#e' = + 6"-37 sin (2* - # ) 



- 2"-52 cos 2 (2* - $ ) 



- l"-l6 sin3(2*-$). 



e'Xv'm- 5"-94sin (2*-$)- l"-36 cos (2*-#) 



- 3"-86 sin 2 (2* - # ) + 4"-12 cos 2 (2* - $ ) 



+ 2"'32cos3(2^ -#). 



^'n = + 0"-0006259 sin (2* - $ ) + 0"-0002319 cos (2* - # ) 

 + 0"-0001526 sin 2 (2* - Ig ) - 0" '0000808 cos 2 (2* - $ ) 

 + 0''-0000180 sin 3(2^ - $) - 0" "0000291 cos 3 (2* - ig). 



Xri = - 0"-0004608 sin (2* - $ ) - 0"-000l685 cos (2^ - ^ ) 



- 0"-0001113 sin 2 (2* - W) + 0"-0000587 cos 2 (2^ - $) 



- 0"'0000132 sin 3(2* - $) + 0" -0000212 cos 3 (2* - #)• 



The values of e, e multiplying S'w, §'■&' in the above expressions, are the uncorrected 

 values given in Arts. (5) and (6). 



SECTION III. 



GENERAL EXPLANATION OF THE MODE IN WHICH THE DISTURBING FORCES 

 PRODUCE THE LONG INEQUALITY. 



62. In attempting to explain the mode in which the long inequality is produced, we 

 shall first briefly describe the nature of the disturbing forces when both orbits are supposed 

 to be circular and the periodic time of the exterior planet double, or nearly double, of that of 

 the interior. Then supposing each orbit in turn to become elliptical while the other remains 

 circular, we shall shew how the alteration in the planet's place, and in the direction of its path 

 due to the elliptic motion, gives rise to small additional disturbing forces which do not go 



