23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 



P.S. In the Compte Rendu of April 9, I860, which has appeared since 

 the foregoing paper was read, M. de Pontecoulant gives the value of the 

 secular acceleration of the Moon's mean motion, which he has obtained by 

 taking the time as the independent variable, and which he considers to be 

 " desormais a 1'abri de toute objection." 



This result, however, of M. de Pontecoulant's is the same as that which 

 he formerly communicated to me, the error of which I have already pointed 

 out. 



M. de Pontecoulant thus describes his method, " En developpant la 

 formule qui donne 1'expression de la longitude vraie en fonction de la lon- 

 gitude moyenne, et en n'ayant egard qu'au premier terme de ce developpe- 

 ment, c'est-a-dire a sa partie non-periodique j'en ai conclu le rapport du 

 moyen mouvement de la lune dans son orbite troublee au moyen mouvement 

 relatif h son orbite elliptique, c'est-a-dire a 1' orbite que cet astre decrirait 

 autour de la terre sans 1'action du soleil... En differentiant ensuite cette 

 valeur par rapport a 1'excentricite e' de 1'orbite terrestre qu'elle renferme,... 

 j'ai obtenu une expression de cette forme : 



^- = #8.6*." 



n 



The value of H thus obtained is 



3 5337 



which, as I have shewn in p. 9 (see p. 167 above), is the result that would be 

 found by differentiating the relation between n and a, and then neglecting 

 the variation of a. The fallacy of M. de Pontecoulant's reasoning consists in 

 his treating the Moon's " orbite elliptique, c'est-a-dire, 1'orbite que cet astre 

 de'crirait autour de la terre sans 1'action du soleil," as if it were a real 

 elliptic orbit with an unalterable semi-axis major, whereas the semi-axis 

 major of the elliptic orbit spoken of by M. Ponte~coulant, which is the same 

 quantity as that above denoted by the symbol a, is really variable, and its 

 variation must be found by means of the differential equations in the way 

 which I have before described. 



The numerical value of the coefficient of the secular equation which 

 M. de Pontecoulant obtains in this paper, when reduced so as to correspond 



with the value -1270'Y of the integral ((eP-E^ndt is 7 "' 96 which ' as 



232 



