66 LECTURES ON THE LUNAR THEORY. [LECT. 



In the actual case considered we notice that q a does not differ widely 

 from unity. Hence k is nearly equal to unity also, and the denominator 

 in c_j is small, and makes c_ l much more important than c,. 



If we substitute these values in the third equation above, we have 



whence 



(# _ gr o )3 _ 8 (# _ % )= _ { 1 6 ( tfo _ i ) + 2?I 2 } (* - ) - 8 Sl s = 0, 



which may be put under the form 



4 8 



whence 



/ 7 2 i \ " / i \ " 2 / 7 ** \ i / 7 Vt 



4 8 



With this equation we can approximate very rapidly to the value of k. 

 Taking as a first approximation 



I- - <la = 0, 



substitute this value of k in the small terms and we get as a second 

 approximation 



k= 1-08516,9. 



Whence the ratio of the retrograde motion of the node to the Moon's mean 

 motion is 



k/n-1 =g-l= -00399,7, 



where g is written for k/n. This value is very correct. Taking the Moon's 

 mean annual motion as 17325593", the resulting annual retrograde motion 

 of the node is 



69252" = 19 14' 12". 



Next find the values of the coefficients c_,, c 1( c_ 2 , c 2 . We have 



<?,= -01261,5, 

 q a -(k-2Y= -34112,3, 

 q -(k + 2)-= -8-34022,8; 

 whence as a first approximation 



c_!= -'03698,19, c 1 = -00151,26. 



