E-3 The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution) 559
Continuation of Box.
1
λ 2 = ρ
2 [1 + c 02 l
2 + c 12 bl
2 + c 04 l
4 + c 22 b
2 l
2 + c 14 bl
4 + c 32 b
3 l
2 + O λ 2 (b
6 , l
6 )] ,
λ
2 =
1
ρ 2 [1 + d 02 (q
2 )
2 + d 12 q
1 (q
2 )
2 + d 22 (q
1 )
2 (q
2 )
2 +
+d 32 (q
1 )
3 (q
2 )
2 + d 04 (q
2 )
4 + d 14 q
1 (q
2 )
4 + O λ 2 ((q
1 )
6 , (q
2 )
6 )] ,
(E.77)
ds
2 = λ
2 (q
1 , q
2 )
ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.78)
(λ
2 (q
1 , q
2 ) from series inversion of q
1 (l, b) and q
2 (l, b) leading to l(q
1 , q
2 ) and b(q
1 , q
2 ).)
The coefficients c µν and d µν are defined as follows
(η
2
0 =
e
2
1−e 2 cos
2 Φ 0 and t = tan Φ 0 ):
c 02 = (1 + η
2
0 ) cos
2 Φ 0 ,
c 12 = −2t 0 (1 + 2η
2
0 ) cos
2 Φ 0 ,
c 04 =
1
12
(8 − 4t
2
0 + 20 η
2
0 − 28 η
2
0 t
2
0 + 16η
4
0 − 48η
4
0 t
2
0 − 24η
6
0 t
2
0 + 4η
6
0 ) cos
4 Φ 0 ,
c 22 = (t
2
0 − 1 − 2 η
2
0 + 6 η
2
0 t
2
0 ) cos
2 Φ 0 ,
c 14 =
1
3
(−7 + 2t
2
0 − 44η
2
0 + 28 η
2
0 t
2
0 − 56η
4
0 + 72η
4
0 t
2
0 + 48η
6
0 t
2
0 − 22η
6
0 )t 0 cos
4 Φ 0 ,
c 32 =
2
3
(2 + 10η
2
0 + 6 η
2
0 t
2
0 )t 0 cos
2 Φ 0 ,
(E.79)
d 02 =
−2(1 − e
2 sin
2 Φ 0 )
2
A 2 (1 − e 2 )
,
d 12 =
4(1 − e
2 sin
2 Φ 0 )
5/2
A 3 (1 − e 2 ) 2
sin Φ 0 cos Φ 0 ,
d 22 =
−2e
2 (1 − e
2 sin
2 Φ 0 )
3
A 4 (1 − e 2 ) 3
×
×[−1 + 2 sin
2 Φ 0 + e
2 sin
2 Φ 0 (6 − 7 sin
2 Φ 0 )] ,
d 32 =
−4e
2 (1 − e
2 sin
2 Φ 0 )
7/2
3A 5 (1 − e 2 ) 4
×
×[2 + e
2 (9 − 22 sin
2 Φ 0 ) + e
4 sin
2 Φ 0 (−24 + 35 sin
2 Φ 0 )] sin Φ 0 cos 0 ,
d 04 =
3(1 − e
2 sin
2 Φ 0 )
4
A 4 (1 − e 2 ) 2
−
(1 − e
2 sin
2 Φ 0 )
3
12A 4 (1 − e 2 ) 3 ×
×[1 + e
2 (3 − 5 sin
2 Φ 0 ) + e
4 sin
2 Φ 0 (−27 + 28 sin
2 Φ 0 )] ,
d 14 =
−12(1 − e
2 sin
2 Φ 0 )
9/2
A 5 (1 − e 2 ) 3
sin Φ 0 cos Φ 0 +
2e
2 (1 − e
2 sin
2 Φ 0 )
7/2
A 5 (1 − e 2 ) 4
×
×[2 + e
2 (9 − 19 sin
2 Φ 0 ) + e
4 sin
2 Φ 0 (−27 + 35 sin
2 Φ 0 )] sin Φ 0 cos Φ 0 .
(E.80)