27 



traced. The iulegration of the principal equation is usually com- 

 menced by supposing a formula which depends on the knowledge 

 of that integration or on the result of observations. Thus in 

 the Theory of the Moon by M. Laplace, he has, * u being the 

 reciprocal of the Moon's Distance from the Earth, 



and remarks 



" Cette valeur de u suppose Tellipse lunaire immobile ; mais 

 on verra bientot qu'en vertu de Taction du soleil, les noeuds if le 

 perigee de cette elHpse sont en mouvement. Alors, en designanl 

 par (1 — c) c le mouvement direct du perigee, &c. 



« = h--(!-ty^ ] ^ ' + 4 r- + e cos (c— t) &c.^ 

 This is his first approximation, but certainly the first approxima- 

 tion should be the former, and the second should be deduced 

 therefrom by a regular process. The result undoubtedly con- 

 firms this hypothesis, but it seems more consonant to the usual 

 steps of mathematical reasoning to deduce one from the other, 

 this is an object in the following investigation, in which also 

 the meati motion of the perigee is computed by confining the 

 process principally to this point, and therefore will be easily 

 intelligible to those who may be unwilling to encounter the 

 formidable calculations necessary for the complete lunar theory. 

 The method of integration which is applied to the differen- 

 tial equation of the lunar orbit is peculiarly convenient for 

 the above purposes. With respect to this method it does not seem 

 necessary here to remark more respecting it, than only to men- 

 tion, that it principally derives its convenience from certain theo- 



F 2 

 * Mec. eel. p. 187. Liv. 7. Tom. 3. 



