21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 153 



20. The existence of the new terms in the expressions for the Moon's 

 coordinates occurred to me some time since, when I was engaged in thinking 

 over a new method of treating the lunar theory, though I did not then 

 perceive their important bearing on the value of the secular equation. 



My attention was first directed to this latter subject while endeavouring 

 to supply an omission in the theory of the Moon given by Pontecoulant 

 in his Theorie Analytique. In this valuable work, the author, following 

 the example originally set by Sir J. Lubbock in his Tracts on the Lunar 

 Theory, obtains directly the expressions for the Moon's coordinates in terms 

 of the time, which are found in Plana's theory by means of the reversion 

 of series. With respect to the secular acceleration of the mean motion, 

 however, Pontecoulant unfortunately adopts Plana's result without exami- 

 nation. On performing the calculation requisite to complete this part of 

 the theory, I was surprised to find that the second term of the expression 

 for the secular acceleration thus obtained, not only differed totally in mag- 

 nitude from the corresponding term given by Plana, but was even of a 

 contrary sign. My previous researches, however, immediately led me to 

 suspect what was the origin of this discordance, and when both processes 

 were corrected by taking into account the new terms whose existence I 

 had already recognized, I had the satisfaction of finding a perfect agree- 

 ment between the results. 



A. 



20 



