98 J. N. Stockwell— Researches on the Lunar Theory. 
avoided. However, in order to show the rapid convergency of 
the series which determine the principal periodic inequalities 
depending on the eccentricity and inclination of the orbit, I 
here give the two terms of the coefficient of the evection pee 
I have computed. The first two terms laa: on the first 
power of the eccentricity are as follows 
4280"-9-+4+122"°0, 
while Delaunay gives the following terms: 
31767°4+-10417°54-297"°54+72"'3. 
It is evident that the first series converges about ten times 
as rapidly as the secon 
The gone comparison is sufficient to show the correct- 
ness bac value of th e method which I have employed in the 
problem of ant moon’s motion; and I shall now mention a few 
cases in which my results are wholly different from what other 
calculators have found for the same inequalities. 
Before doing so, however, I would observe that there are 
certain fundamental and axiomatic conditions which ou ught to 
be satisfied by the results arrived at, whatever be the method 
of analysis which we may employ. In the present case the 
condition to be satisfied is simply, That all the terms introduced 
into the expressions of the codrdinates by the disturbing function 
ought to disappear when the disturbing function is pul e equal to 
nothing. It is, however, a remarkable fact in connection with 
the lunar theory, that, among the four hundred and seventy-nine 
equations of the longitude given by Delaunay, there are /ive, 
arising from the sun’s attraction, which do not disappear when 
the disturbing function is put ‘equal to nothing. From this 
circumstance it is easy to conclude that there must be some- 
thing seriously wrong in his development, notwithstanding its 
intricacy and refinement. The same remark is also applicable 
to the lunar theories of LaPlace, Plana and Pontécoulant. 
The most important of these equations are those having the 
arguments, 2¥ — J, and D +’, in Delaunay’s theory ; or, twice 
> moon's distance Jrom the node minus the mean anomaly, an 
the moon's longitude minus the longitude of the sun’s perigee 
_ The first of these is an inequality of pure elliptic motion, with 
a coefficient of +45’-4, while the coefficient arising from per- 
turbation amounts to gd” 8, 28” only of which disappears 
when the disturbing function is put equal to nothing. Accord- 
ing to my analysis, the coefficient of this inequality arising 
from perturbation amounts to only 0’”"18, a quantity less than 
a four — part of Delaunay’s coefficient arising from the 
same ¢ 
he ‘qveticiett of the second equation, mentioned above, 
