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dicult, we find
∆
a
=0
,
e
2
−
1
∆
e
=
−
tan
i
∆
i,
e
∆
i
=3
πe
a
e
p
3
1
sin
2
i
cos
i
cos
ωJ
3
5
4
−
16
πe
a
e
4
1
sin
2
i
sin 2
i
sin 2
ωeJ
4
+
45
7
6
−
···
,
p
3
π
a
e
p
2
∆Ω =
−
cos
iJ
2
+3
π
a
e
p
3
1
−
sin
2
i
cot
i
sin
ωeJ
3
15
4
(7-23)
π
a
e
p
4
1
sin
2
i
cos
iJ
4
+
15
2
7
4
−
···
,
∆
ω
=6
π
a
e
p
2
1
sin
2
i
J
2
5
4
−
+3
π
a
e
p
3
1
sin
2
i
sin
i
sin
ωeJ
3
5
4
−
4
1
−
−
15
π
a
e
p
sin
4
i
31
8
sin
2
i
+
49
19
sin
2
i
sin
2
i
cos 2
ω
J
4
+
3
7
16
8
−
···
.
Terms of the order of
e
2
J
3
and
e
2
J
4
, which are very small, have been ne-
glected in these equations. The proportionality of ∆
e
and ∆
i
is more or less
accidental: it applies only with respect to long-periodic disturbances;
e
and
di/dt
themselves are not proportional. The quantity
p
is defined by (7-6);
it is hardly necessary to repeat that
a, p, e
, etc., refer to the orbital ellipse
and not to the terrestrial ellipsoid, of which
a
e
is the equatorial radius.
By integrating over one revolution, we have removed the
short-periodic
terms of periods
P,
2
P,
3
P, ...
,suchascos
v,
cos 2
v
,etc.Whatremainsare
secular
terms, which are constant for one revolution and increase steadily
with the number of revolutions, and the
long-periodic
terms, which change
very slowly with time in a periodic manner. The argument of perigee
ω
increases slowly but steadily, so that the perigee of a satellite orbit also
