271 



force, are not, in a mathematical sense, legitimate processes ; and 

 that, in the planetary theory, they produce results practically true 

 only on account of the minuteness of the disturbing forces, and the 

 consequent great length of the secular periods ; and that, in the 

 lunar theory, their failure is made evident, in consequence of the 

 comparatively large magnitude of the disturbing force, and the con- 

 sequent rapidity with which the elements of the moon's orbit pass 

 through their secular periods. 



The formulae for the variations of the elements are then applied 

 to the lunar theory ; and some of the integrations are effected 

 by means of a lemma containing the solution of the differential 

 equation 



(where F, p and q are numerical coefficients), in the form 



sM; g 2 F 2 * 



M being (1- -r) 



By this method, the total motion of the moon's perigee, as well as 

 the coefficients of the evectiori, are fully obtained in the first in- 

 stance, without the necessity of any second approximation ; and the 

 usual difficulty as to the movement of the perigee does not present 

 itself. The motion of the node, and the evection in latitude, are 

 correctly obtained in a similar manner. 



This part of the memoir is concluded by an extension of the 

 general formulae for the tangential variation of elements to the case 

 in which we suppose the constant p to become variable, the result 

 being to add to each variation a term involving fye. 



The third part of the Paper contains the development of the 

 method of osculating variation, before briefly described ; from which 

 are deduced the formulae for the osculating variations of elliptic ele- 

 ments. This method is capable of being applied to the planetary 

 and lunar theories, as well as that of tangential variation ; but the 

 advantages of this method did not appear to be such as to justify 

 the actual expansion of the formulae for these theories. The author, 

 however, shows that with reference to any system of three bodies, 

 the equations of motion for each body naturally assume the form 



#" + ra 2 0-X) = 0, &c. 

 (being the system solved by this method) ; and that the X, Y, and 



VOL. IX. \j 



