OF URANUS AND NEPTUNE. XXXI 



38. We may now collect together the variations of the elements, so far as they depend 

 upon the first power of the disturbing force. Putting fJJ = mean long, of Uranus, * = mean 

 long, of Neptune, and therefore writing $ - 2* for X, the variations of the elements of 

 Uranus will be 



S (nt + e) = + 2783"-825 sin (2* - $ ) + 42l"-244 cos (2* - # ) 

 + 125"-0l6 sin2(2*-]g ) + 3"-6n cos 2(2*-$) 

 + 8"' 148 sin 3 (2* - $ ) + 0"'436 cos 3(3* - $ ). 



le m - 85"'105 sin (2* - # ) + 408"'S50 cos (2* - # ) 

 6"-989 sin 2 (2* - # ) + 32"-541 cos 2 (2* - # ) 

 0"-759 sin 3 (2* - # ) + 3"-219 cos 3(2* - $ ). 



e$vr = + 409"*040 sin (2* - $) + 84"-810 cos (2* - $) 

 + 32"-498 sin 2 (2* - i$) + 6"-988 cos 2 (2* - $) 

 + 3" -219 sin 3 (2* - $) + 0"-759 cos 3 (2* - $). 



Sn m - 0"-00184260 sin (2* - $) + 0"-01228015 COS (2*- Ig) 



- 0"-00003752 sin 2 (2* - $) + 0"-001 13009 cos 2 (2* - ]g) 



- O''-OO0O0689 sin 3(2* - $) + 0"*00011349 cos 3(2* - $). 



Sa = + -00055795 sin (2* - $) - -00371851 cos (2* - $) 

 + -00001136 sin 2(2* - $) - "00034220 cos 2(2* - $) 

 + -00000209 sin 3(2* - $) - -00003436 COS 3(2* - $). 



&y = - 0"-582 sin (2* - $) + l"-146 cos (2* - Ig) 



- l"-503 sin 2 (2* - $ ) + l"-353 COS 2 (2* - Ig.) 



- 0"-2l6 sin3(2*-^[) + 0"*195 cos3(2*-^). 



SlI = - 77"'890 sin (2* - # ) + ll"-477 cos (2* - ¥ ) 

 + 49"-760 sin 2 (2* - #) + 56"'954 cos 2 (2* - # ) 

 + 7" -415 sin 3(2* - #) + 8"-231 cos 3 (2* - # ) ; 



and the variations of the elements of Neptune, are 



3 (»'* + e') = - 1996"-662 sin (2* - $ ) - 302"-139 cos (2* - $ ) 

 - 89"-358 sin 2 (2* - # ) - 2" -599 cos 2 (2* - $ ) 

 5"-803 sin 3 (2* - $ ) - o"'313 COS 3 (2* - $ ). 



&>' * + 89"-999 sin (2* - $ ) + 83" -276 cos (2* - # ) 

 + 24" -981 sin 2 (2* - # ) + 23"'490 COS 2(2* - # ) 

 + 2"-451 sin 3 (2* - # ) + 2"-406 cos 3(2* - # ). 



e'5w' = + 83"-024 ?in (2* - # ) - 90" - 762 cos (2* - $ ) 

 + 23"-501 sin 2 (2* - # ) - 24"-981 cos 2 (2* - # ) 

 + 2"-406 sin 3 (2* - # ) - 2"-451 COS 3 (2* - $ ). 



