465]
NOTE ON THE LUNAR THEORY.
363
Plana, with Elliptic e, y
Elliptic
y= y =
7 - 7e 2 -§7 3
+
<
Cb
1
-It 3
sin
9
+ ye — 7e 3 — § y 3 e
+ ye — 1 ye 3
-f 7 3 6
33
c+ g
+ ye — § 7e 3 + -} fe
+ ye
-f 7 3 e
33
0- g
+ f 7e 2
+ î 7e 2
33
2c- g
+ 1 7e 2
+ 1 7e 2
33
2 c + g
+ Ü7*
+ t 3 2 7¿ 5
33
Sc- g
+ 1 7e 3
+ 1 ye 3
33
Sc+ g
-^T 3
“A 7 3
33
Sg
+ i y 3 e
- £ fe
33
c-Sg
- h fe
- i 7 3 e
33
c + Sg,
where, for greater clearness, I remark that the values called “ elliptic ” of e, y, c, g,
refer to an ellipse, such that the longitude of the node, and the longitude (in orbit)
of the pericentre, vary uniformly with the time,—viz., we have mean distance = 1,
excentricity — e, tangent of inclination = 7, mean longitude = l, mean anomaly = c,
distance from node =g.
We have therefore
8- =
r
- f 7 2 ^ 2
COS
— f 7 2 e
55
c —2g
Sv =
- Te 7 2 e 2
sin
2c
>3
*9
+ f 7 2 6
35
c —2g
- te 7 2 e 2
3?
2c — 2g
Sy =
- f 7e 3 +1 7 3 e
33
c- g
+ t 7e 2
33
2c- 9
+ f 7e 3
33
Sc- g
+ f 7 3 6
33
c-Sg,
viz., these are the increments to be added to the elliptic values of -, v, y, respectively,
in order to obtain the disturbed values of v, y, attending only to the coefficients
independent of m; they represent, in fact, the lunar inequalities which rise two orders by
integration.
The elliptic values of ^ and y are functions, and that of v, is equal l +, a function,
of e, 7, c, g, and the foregoing disturbed values may be obtained by affecting each of
46—2