100 JS. N. Stockwell— Researches on the Lunar Theory. 
and node of the lunar orbit, and the difference of longitudes of 
perigee of sun and moon, respectively. Plana was the firs 
give a correct approximate solution of the problem of the first 
of these inequalities, which is produced wholly by the varia- 
tions of the central force. By means of a laborious investiga- 
tion, occupying about fifty pages of his Theory of the Moon’s 
Motion, he has obtained a tolerably scare approximation to 
the value of the inequality. He obtains +1/°405 for the sum 
or the elliptic and perturbed Goatiescnk: but the elliptic co- 
efficient is equal to —0/’-932; whence it follows that the coefii- 
cient due to perturbation amounts to about +2’34. I obtain, 
almost without labor, +2’54 for the value of this coefficient. 
The second inequality is produced by both classes of forces, 
and the determination of its coefficient is more complicated 
than that of the inequality just mentioned. The value of the 
force of class (A), which produces the inequality, is about one- 
jifth of the former, but it has a period about three times as long. 
The inequality produced by this force ought to be about three- 
Jijths of the former inequality, which would make it equal to 
1’’52. But the tangential force is far more effective, since the 
inequalities produced are proportioaal to the squares of the 
pow of the arguments. I find, however, by an exact calcu- 
tion that the part of the coefficient of this inequality which 
arises from the central force amounts to +1/°45; while the part 
of it which arises from tangential seis amounts to +107’08; 
thus making the coefficient of the inequality equal to 108’’53. 
The solutions of Plana, Pontécoulant and Delaunay, all make 
the coefficient equal to about 0’”4, when quantities of the same 
order only are include 
It is remarkable that the inequalities of long period arising 
from the two classes of forces which produce them should 
follow the same law as the acquired velocity, ave space peer 
over, by falling bodies at the surface of the earth, the on 
being. De proportional to the time and the other the square of 
e t 
Tf we extend the comparison to the variation of the ele- 
ments, we shall find that the method which I have employed 
possesses the advantage of more rapid convergency. For 
example, I find for the first two terms of the mean motion of 
the perigee the following value 
0°00419643 +- 0° 00395575 = 9°00815218, 
while the first three terms of Delaunay’s series are 
0°00419643 + 0°00294279 + 0°00099570 = 000813492. 
This comparison shows that two terms of my series are con- 
siderably more accurate than three terms of Delaunay’s. 
The preceding comparisons are sufficient to databligh two 
