232 
MR. LUBBOCK’S RESEARCHES 
are retained, may be seen in his invaluable memoir. Mr. Brice Bronwin has 
recently communicated to the Society a lunar theory, in which the same 
method is adopted. 
Having reflected much upon the difficulties of this problem, I am led to 
believe that the integration of the differential equations in which the time is 
the independent variable, is at least as easy as the method hitherto, I think, 
solely employed, and I now have the honour to submit to the Society a lunar 
theory founded upon this integration, which is in fact merely an extension 
of the equations given in my Researches in Physical Astronomy, already printed, 
by embracing those terms which, in consequence of the magnitude of the 
eccentricity of the moon’s orbit, are sensible ; and the suppression of those, 
on the other hand, which are insensible on account of the great distance of the 
sun, the disturbing body. By means of the Table which I have given (Table II.), 
the developments may all be effected at once with the greatest facility. 
The first column contains the indices, which I have employed to distinguish 
the inequalities. The numbers in the second column are the indices affixed 
by M. Damoiseau, in the Mem. sur la Theor. de la Lune, p. 547. to the inequa- 
lities of longitude. 
t* — nt — n t t, x = cnt — vr, z — n t t — y — gnt — v. 
0 
0 
21 
45 
2 t — 3 x 
42 
73 
2 t — 3 x — 2 
1 
30 
2 1 f 
22 
46 
2t + 3x 
43 
. . 
2 t + 3 x + 2 
2 
1 
X 
23 
21 
2 x + z 
44 
26 
3 x — z 
3 
31 
2 t — x \ 
24 
53 
2 t — 2 x — z 
45 
2 t — 3 x + 2 
4 
32 
2 t q- x 
25 
54 
2 l -f- 2 x -f- z 
46 
2 t + 3 x — 2 
5 
16 
z § 
26 
20 
2 X — 2 
47 
2x + 2 2 
6 
33 
2 t — z 
27 
51 
2 t — 2 x -f 2 
48 
75 
2 t — 2x — 2z 
7 
34 
2 t -\- z 
28 
52 
2 t + 2 x — 2 
49 
2t + 2x + 2z 
8 
2 
2x 
29 
23 
X + 22 
50 
2 x — 2 z 
9 
35 
2 t — 2 x 
30 
59 
2 t — x — 2z 
51 
2 t — 2 x + 2 2 
10 
36 
2 t + 2x 
31 
, , 
2 t -\- x + 2 z 
52 
2 f + 2 x — 2 2 
1 1 
19 
X + z 
32 
22 
X — 2 2 
53 
x -j- 3z 
12 
41 
2 t — x — z 
33 
61 
2 t — x + 2 2 
54 
2 t — x — 3 2 
13 
42 
2 t + x -f z 
34 
60 
2 t + x — 2 z 
55 
2 t + x + 3 z 
14 
18 
X — z 
35 
. 
3 z 
56 
x — 3 z 
15 
39 
2 t — x + z 
36 
2 1 — 3z 
57 
2 t — x + 3 2 
16 
40 
2 t -f- X — 2 
37 
2t + 3 2 
58 
2 t + x — 3 2 
17 
17 
2 2 
38 
9 
4 x 
59 
42 
18 
43 
2 t — 2z 
39 
67 
2 t — 4 x 
60 
2 t — 4z 
19 
44 
2 t + 22 
40 
2 t + 4 x 
61 
2 t + 42 
20 
4 
3 x 
41 
27 
3 1+2 
62 
3 
2 y 
* Inconvenience arises from using the letter t in this acceptation. I have done so in order to con- 
form to the notation of M. Damoiseau. + Variation. + Evection. § Annual Equation. 
