We thus come to the conclusion that we shall reach a sufticient 
approximation by simply superposing the fields of the earth and 
of the sun, both computed as if these bodies were at rest, i.e. by 
taking throughout ’) 
dij = (ido + (Yi jr’: 
Then we have rigorously 
d* a; ie ER d* xj ‘Si 
EN rek Wire = Aj 2 a ati 
dt dt 1 dt 0 dt 0 
where the meaning of the different suffixes will be easily under- 
stood. The first term gives the same result that has already been 
found above for the planetary motion, viz. a secular motion of the 
perigee amounting to — —.xnt. In one century this is 0°.06, which is 
P 
entirely negligible. The other terms are found by writing the equation 
(20) for the moon and for the earth and subtracting. the latter from 
the former. The left-hand members then give the perturbing function 
as used in the current lunar theory. This contains the factor #m/‚5, 
which is by KerPrer’s third law replaced by n’, 7’ being the mean 
motion of the sun. Now however this law must be replaced by (34) 
and we have consequently 
k?m iS 3h? 
ee 1 + ——}. 
o° o 
We must therefore apply a correction to the ordinary perturbing 
function. ; 
The right-hand members of (20) give 
ax; 4)°x; AME; 4xiÂ 45; 0 x; Cie on ; 
Jd — = k?m -— — ———-—-—_ 7 + —g,7;, (96) 
dt? aN o° ES O° Ds 0 
where we have put 
Oo aS ti ; (Des (> from 1 to 3). 
Further we have x;= 6 + a; ete., and 
We develop in powers of "/, and we neglect the square and higher 
powers of this small quantity. We also take, as has already been 
remarked, o = 0; and we neglect all periodic terms. I then find 
the following radial, transversal and orthogonal perturbing forces 
1) Simuitaneously with the present communication Mr. Drosre has published 
(these Proceedings, June 1916) the complete values of gij for m moving bodies. 
His results applied to the lunar theory entirely confirm the conclusion which was 
reached above. 
