178 SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [23 



to put his finger on the error I have committed. From an equation in 

 my Memoir he deduces the following : 



e' = q + q' -lv m 2 sin (2v 2mv) m?e' sin 2 (2r 2rnv c'mv) + &c. r 



[8 lb J 



and then adds the remark, 



" C'est-a-dire, que 1'exceritricite de 1'orbite terrestre, outre sa variation 

 seculaire, serait soumise a toutes les indgalites du mouvement lunaire ; c'est- 

 a-dire, a des variations dont le periode serait d'un mois, d'une annee, &c. 

 ce qui est contraire, quelque petitesse qu'on suppose au coefficient q', k tous 

 les principes de la theorie." 



Now it is astonishing that M. de Pontecoulant does not see that the 

 quantity enclosed within brackets, in the above equation, is simply the 

 expression of the Moon's mean longitude nt in terms of the true longitude 

 v, so that the equation is equivalent to 



c' = q + q'nt ; 



that is, the eccentricity of the Earth's orbit is made to vary uniformly with 

 the time, which agrees with the supposition with which we started. 



On the other hand, M. de Pontecoulant, by making 



e'=q+ q'v, 



that is, by supposing the change in e' to be proportional to the Moon's 

 true motion in longitude, would evidently cause the eccentricity of the 

 Earth's orbit to be affected by all the inequalities of the lunar motion. 



All attempts to express e' in terms of v, without introducing periodic 

 terms, lead to this absurdity. 



I have already alluded to the strange notion expressed at the end of 

 M. de Pontecoulant's paper, that there may be two values of the secular 

 acceleration, one applicable to the true longitude and the other to the 

 mean longitude. The difference between the true and the mean longitudes 

 consists wholly of periodic quantities, and cannot contain any term increasing 

 continually with the time. 



How M. de Pontecoulant could have so far deceived himself as to imagine 

 that this paper settled the question of the secular acceleration, " sans con- 

 testation possible desormais," is, I confess, beyond my comprehension. 



