J. N. Stockwell—Inequalities in the Moon’s Motion. 9 
[5376] which he afterwards uses in his reductions. However, 
as I have shown above that equation [5372] is the same as 
[5362], and has a contrary sign, it is unnecessary to pursue this 
part of the inquiry further, since it is evident that the whole 
value of dv must be derived from the value of F in [5362]. 
LaPlace has given the complete value of dédv corresponding 
to the plane of the orbit, in [5379]; and he gives a correction 
in [5885] to reduce it to the plane of the ecliptic. It is appar- 
ent, however, that this correction does not exist, for LaPlace 
has shown in [928] etc., where this subject is first investigated, 
that this correction is of the order of the square of the disturb- 
ing force; and as terms of that otder Seve not been con- 
sidered, it is evident that the value of that correction which he 
has given in [5885] is erroneous. 
To complete this subject, it now remains to be shown that 
the value of 2 in equation [5362] gives the value of dév twice 
as great as LaPlace has found in [5368]. For this purpose I 
would remark that the value of dév given by means of [5367], 
1s the correction to the disturbed mean longitude, and not to the 
undisturbed mean longitude. In order to correct for this 
condition it is necessary to add the term 3 ei [rr to the first 
rdv 
member, and this cancels the same term in the second member, 
thus leaving the correction to the wndisturbed mean longitude, 
al 
or dv equ to 
9 a *) 
facet “dr | 
This will be apparent from the considerations given in § 54 of 
Book IL of Mécanique Céleste, from which it appears that the 
function dR has a term of the form sin (at + ), in which ais 
very small, and gives by a double integration a* as a divisor; 
and LaPlace has shown in [1070’] that for this case we must 
increase the mean longitude by the quantity Bh: f nat J ak, 
# 
which i ‘dt! } h 
is equal to af SS [ik or to the first term of the 
second member of equation [5367]. It therefore follows that 
the complete value of dav will be given by the equation 
d@ (dR | 
ddv = 2 ade” (=)- (35) 
and if we substitute the value of (@) given by [53865] it 
: i 
becomes equal to twice equation [5368], or identically the same 
as I have obtained by an entirely different method. 
Cleveland, Ohio, Oct. 29, 1879. 
