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



give rise to the following parts of the coefficient of the secular equation : 



m 2 10-66, 



m 4 - 2-34, 



m 5 - 1-58, 



m 6 - 0-71, 



TO 7 - 0-25. 



The sum of these is 5"'78. The convergence, although slow at starting, 

 becomes more rapid in the later terms ; and I inferred, in my communi- 

 cation to the French Institute above mentioned, that the remainder of the 

 series would be very nearly equal to 0"'08. 



Now M. Delaunay has since calculated the next term of the series, 

 and finds it = 0"'06, which is in exact accordance with my anticipations. 



Although I think that there can remain no doubt with respect to the 

 convergency of the series, yet, in order to remove all possible objection, I 

 have calculated the value of K by a method which does not require any 

 expansion in powers of m, and the resulting coefficient of the secular equation 

 is 5 //- 70, exactly agreeing with that found by means of the series of powers 

 of m. 



A very few words will now suffice in reply to the objections which 

 M. de Pontecoulant brings forward in his observations in the Monthly 

 Notices. In fact, almost all of them have been virtually answered in what 

 I have said before. 



At the outset of his paper, M. de Pontecoulant rightly describes the 

 difference between my method of finding the secular acceleration and all 

 preceding ones, as arising from the consideration of the variability of the 

 eccentricity of the Earth's orbit in the differential equations of the Moon's 

 motion, in which this element had hitherto been considered as constant. 

 He then refers to the statement in my Memoir, that when this consideration 

 was introduced into the formulae, I found exactly the same result whether 

 the time or the Moon's longitude was taken as the independent variable. 

 But, adds M. de Pontdcoulant, "il n'y a qu'une petite difficult^ dans cette 

 assertion, c'est qu'elle enonce un fait mathematiquement inadmissible." 



Now I confess that I cannot see M. de Ponte"coulant's "petite difficult^." 

 I am far from looking upon the agreement between the results of different 



