24] IN A PARTICULAR CASE. 185 



this case, if /A denote the sum of the masses of the Earth and Moon, we 

 shall have 



The differential equation which determines z, the Moon's coordinate per- 

 pendicular to the Ecliptic, is 



d"z 



Now, the Sun's orbit being circular, we have ^ = n'\ and the only 



function of the Moon's coordinates which we require in order to form this 



1 

 equation is -^. 



I find that, with the above unit of distance, 

 ~= 1-00280,21783,115 



+ 0-021 59,98364,4 cos 2(n-n')t 



+ 0-00021, 53273,9 cos 4(n-n')t 



+ 0-00000,20644,8 cos 6 (n - n'} t 



+ 0-00000,00192,9 cos 8(n-n')t 



+ 0-00000,00000,3 cos 10 (n-n r ) t. 

 Let 



_L_ /_* , 2 \ J_ (l_ 



OF / /\o I "r W , / \, I ., + 1 



(n - nj V T r t V ' (n - nj \r> " J ' (1 -m) \r 



= q- + 2q l cos 2 (n n') t + 2q 2 cos 4 (n n') t + 2q 3 cos 6 (n n')t + &c. ; 

 then we find, from the above value of -^ , that 



2 3 = 1-17804,44973, 149, and q = 1-08537,75828,323, 

 ft = 0-01261,68354,6, 

 2 2 = 0-00012,57764,3, 

 q, = 0-00000,12059,0. 



These are all the quantities necessary for finding the motion of the 

 Moon's node, to the order which we require. 



If gTr denote the angular motion of the Moon from its node in half a 

 synodic period of the Moon, the equation so often referred to above gives 

 A. 24 



