LECT. XIV.] THE LATITUDE AND THE MOTION OF THE NODE. 65 



Let us now consider the equation 



where P = q + 2q 1 cos 2t + 2q* cos 4i, 



in which q lt <, are supposed small. 



Suppose a term in z to be c cos (kt + /8) ; when this is substituted in 

 Pz there will arise terms 



ccos(k-2t + fi) ccos(k + 2t + /3) 



Let us therefore assume 



+ c_i cos (k -2t + /3) + c_, cos (k- it 

 c is arbitrary; we have to determine k, c lt c_ lt &c. 

 Substitute and equate coefficients : 



-0 



,+ ... =0 



q, +\_-(k+2' ! + ~c + c+ ... =0 



...... + q, + qf, 



If q,, q.,, ... are neglected, we have simply 



this is a first approximation to the value of k. 

 Taking q 1 into account and neglecting <? 2 



A. II. 



