6 J. N. Stockwell—Inequalities in the Moon's Motion. 
while LaPlace found 
du =— PE y sin = + 7°03 sinQ. (15) 
If, in the development of the inequalities depending on the 
oblateness of the earth we carry on the approximations so as 
to include terms of a higher order depending’ on the eccentricity 
and inclination of the orbit, we shall find two equations of 
sensible magnitude, having the same arguments as two empiri- 
cal equations discovered by Airy about a third of a century 
ago. ‘These equations depend on the arguments nt — w— Q, 
and nt— w+, in which w denotes the longitude of the 
perigee. These arguments have periods of 27-4432 days, and 
27°6661 days, respectively. The equation depending on the 
first of these arguments seems also to have been independently 
discovered quite recently, as an empirical equation, by Pro- 
fessor Newcomb, who attributes it to the attraction of some of 
the planets. From some calculations which I have made I am 
led to suspect that each of these equations has a value amount- 
ing to quite a large fraction of a second of are; and I call - 
attention to them here as being worthy of a more thorough 
investigation by astronomers. 
t is but proper to add in this connection, that the mean 
motions of the perigee and node of the lunar orbit are affected 
by the oblateness of the earth ; and are also affected by secular 
equations arising from the diminution of the obliquity of the 
ecliptic. The motions of the perigee and node which I have 
obtained agree in value with those obtained by LaPlace. I 
have therefore succeeded in reproducing exactly, by my 
method of computation, all the inequalities in the motions of 
the moon arising from the oblateness of the earth, which 
LaPlace discovered nearly a century ago, with the exception 
of the equation in longitude. The coefficient of LaPlace’s 
equation exceeds the value which I have obtained, in the ratio 
of 19 to 12; and it has been a matter of surprise that two very 
dissimilar methods of computation should give so many results 
identically the same, and leave only a single one discordant. 
This has led me to make a critical examination of every step 
of LaPlace’s calculation of this equation; and this examination 
has developed the fact that LaPlace has, in this instance, 
departed widely from the requirements of his own formule 
and methods; and that a correct calculation by his method 
ke a result identically the same as I have found by my own. 
shall therefore now give the several steps of this examination, 
believing that it will not be without interest to the readers o 
this Journal. In this investigation it has been found con- 
venient to use Bowditch’s translation of the Mécanique Céleste, 
