2 
MR. LUBBOCK’S RESEARCHES 
M. Damoiseau has considered are sufficient, it is unnecessary in either method 
to carry the approximation beyond the fourth power of the eccentricity of the 
Moon, and quantities of that order. 
The method I have employed is equally advantageous in the first approxi- 
mation. I have given in conclusion the numerical results which are obtained 
of the coefficients of the principal inequalities when the square of the disturb- 
ing function is not considered, which may be regarded as an elementary Theory 
of the Moon ; for the differential equations and the equations which serve to 
determine the coefficients retain nearly the same form in the further approxi- 
mations. 
The coefficient of the variation obtained in this manner differs only by a few 
seconds from that given by Newton in the third volume of the Principia; that 
of the erection agrees closely with the value assigned to it by M. Damoiseau. 
This latter agreement of course can only be looked upon as accidental. 
Developments required for the integration of the equation 
d 2 r 2 
d 2 v 3 8 — 
r 
3d 2 
+ 
r +A + 2 / dR+r 
r a 
2d d t* 1 2d t* 
when the square of the disturbing force is retained , 
2/d 
Since r = 1 + ~ — e ^1 — A e 2 ^ cosx — ^1 — "jf ) c os 2 x — ~ cos 3 x — ~ cos 4 x 
[0] 7 [2] [8] [20] [38] 
rt T={( 1+ (‘ -T e 4' s+r< } _ 7{ ,9 + r,0 }} c ° s2 ‘ 
[i] 
+ |(l + -j) r 2--^-( 1 — "jf^) {2r 0 + c 2 r 8 | - e —r 2 jecosx 
[ 2 ] 
+ {( 1 + f) r >-T('-f «■){«% + '.}-? r *}‘ 
•ecos (2 t — x) 
[3] 
+ {(i + ^ ) r4_ y( 1 ~"t e2 ) { r ‘ + e ' r >o}~ Jecos(2l + x) 
M 
COS 2 
[5] 
