6. 



A METHOD OF SOLVING THE EQUATION + Qw = Q, WHERE 



= '+ 2* cos 2< + 2 cos 4>t + 2< cos 



[THIS is the equation which Adams employed to discuss the Lunar 

 Inequalities depending upon the first power of the inclination, and to 

 which Hill reduced the problem of finding those depending on the first 

 power of the eccentricity. From this double claim to the first rank of 

 importance the following method of solving it derives its interest.] 



Two methods of solution have been given already ; firstly it may 

 be solved by evaluation of an infinite determinant as in p. 86 et seqq. ; 

 or again by the method employed in the Lectures on the Lunar Theory, 

 XIV. But in both the important cases to which the equation applies, 

 q is not very different from unity. Hence if we write 



iv c cos (kt + /3) + &c., 



k will also be nearly equal to unity, and it is desirable to find a 

 method which avoids the introduction of the two small quantities q^ k", 

 q ' (k 2) 2 , as divisors. 



Let us assume, omitting c, 



where, in the first place, 



