406 
ON THE SECULAR VARIATION OF THE MOON’S MEAN MOTION. 
19. Transforming the expressions found above, so as to obtain the moon s longitude 
and radius vector in terms of the time, and writing for convenience nt instead of 
mnt instead of and dmnt instead of c!mnt-\-t , we have 
1 1 / 5 \ 74 e^de^ 
sin cos (2-2m)77^ 
, . , dd , , 
— 3?ne sm c mnt — 3^ cos cmnt 
77 215 dd 
+ sin (2 - 2m -- dm)nt-\- cos (2 - 2m - dm)nt 
11 257 dd 
-j^mV sin (2~2m+c'm)^^-^m"^cos (2-2m + cm)a^ 
a , n 4 201 , ,, 
-=aM= 1 “^-m -r^mV 
r 8 16 
cos (2 — 2 m)w^+-p 2 sin ( 2 “- 2 m)«f 
3 
— ^mV cos cmnt — 3^*^ sin dmnf 
-\-^mdd cos (2 — 2m— c'm)w/--^m'^^sin {2 — '2m — dm)nt 
-~\m^d cos (2~2/a+c'm>^+^w'^^sin (2-2m + c'm)«^. 
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 ov'er 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 Analy- 
tique.” 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 unlortunately adopts Plan.v s result M’ithout 
examination. On performing the calculation requisite to complete this part of the 
theory, 1 was surprised to find that the second term of the expression for the secular 
acceleration thus obtained, not only differed totally in magnitude from the correspond- 
ing 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, 1 had the satisfaction of finding a perfect agree- 
ment between the results. 
June \Qth, 1853. 
