144 ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. [21 

 series of small terms involving sines of the same angles, together with a 

 non-periodic part of the form IHe'de' or ^He'-. The introduction of this 



term will evidently change the relation between the non-periodic part of -y- 

 and e' 2 , upon which the secular acceleration depends. 



7. We must commence by finding the new terms to be added to the 

 ordinary expression for u. 



For the sake of simplification we will neglect the eccentricity of the 

 Moon's orbit. 



Let : - denote the non-periodic part of u, and + Su the complete value. 



Then by substitution in the equation for u, making use of Damoiseau's 

 developments of the undisturbed values of the several functions of u, u', 

 and v v which occur in it, putting h- = a /3 and writing, for convenience, 



mv instead of mdv + X, and c'mv instead of c' I mdv + X CT' (as in Plana, 

 vol. I. p. 322), we obtain 



d \a) I 1 d*8u 



Q = TT- + ---- + -3 > + S 

 dv- a a dv 



3 m '* /* / , 3 m ~ i , 3 m- , . , 



o aou' + - - e cos c'mv - {1 + 3e' cos c'mv} aou 



,21m 2 , /r> .,3m 2 . 



+ - - e' cos (2v 2mv c'mv) - - e' cos (2v 2mv + c'mv) 

 4 a, 4 a, 



- \dv\(l-- e h ) sin (2v - 2mv) +-e' sin (2v - 2mv - c'mv) 



a> , i L\ ^ / ^ 



- e' sin (2v 2mv + c'mv) > 



