20-3 Soldner coordinates: geodetic parallel coordinates 435
Continuation of Box.
Second step: determine L, B, γ, starting point P F , s = x = y c , y = x c = 0, α FP = α F0 + 3π/2 = 0 given.
As a result, we compute {u = x, v = 0}, which leads to the series
(the coefficients [µν] are collected in Box 20.5 and have to be computed at the point B F )
B − B F = [20]x
2 + [40]x
4 + · · · (an even polynomial in x) ,
L − L F = [10]x + [30]x
3 + [50]x
5 + · · · (an odd polynomial in x) ,
α PF = γ = [10] α x + [30] α x
3 + · · · ± π (an odd polynomial in x) ,
(20.115)
B = B F + (B − B F ) ,
L = L F + (L − L F ) .
(20.116)
An alternative procedure is the following one-step method: set up a Taylor series for the coefficients
[µν] in (20.115), which have to be computed at the point B F , with B 0 as the point of expansion, i. e.
[µν]| F = [µν]| 0 +
∂[µν]
∂B
˛
˛
˛
˛
0
(B F − B 0 ) +
1
2!
∂
2 [µν]
∂B 2
˛
˛
˛
˛
0
(B F − B 0 )
2 + hit
`
[µν]
´
,
(20.117)
[µν] α | F = [µν] α | 0 +
∂[µν] α
∂B
˛
˛
˛
˛
0
(B F − B 0 ) +
1
2!
∂
2 [µν] α
∂B 2
˛
˛
˛
˛
0
(B F − B 0 )
2 + hit
`
[µν] α
´
.
(20.118)
The terms B F − B 0 , (B F − B 0 )
2 etc. are computed from the first step, for instance,
B F − B 0 = [01]| 0 y + [02]| 0 y
2 + [03]| 0 y
3 + [04]| 0 y
4 + · · · ,
`
B F − B 0
´ 2 =
`
[01]| 0 y + [02]| 0 y
2 + [03]| 0 y
3 + [04]| 0 y
4 + · · ·
´ 2 etc. ,
(20.119)
leading to
[µν]| F = [µν]| 0 +
∂[µν]
∂B
˛
˛
˛
0
( [01]| 0 y + [02]| 0 y
2 + [03]| 0 y
3 + [04]| 0 y
4 + · · ·)+
+
1
2!
∂
2 [µν]
∂B 2
˛
˛
˛
0
( [01]| 0 y + [02]| 0 y
2 + [03]| 0 y
3 + [04]| 0 y
4 + · · ·)
2
+ · · · +
+
1
n!
∂
n [µν]
∂B n
˛
˛
˛
0
( [01]| 0 y + [02]| 0 y
2 + [03]| 0 y
3 + [04]| 0 y
4 + · · ·)
n ,
(20.120)
[µν] α | F = [µν] α | 0 +
∂[µν] α
∂B
˛
˛
˛
0
( [01] α | 0 y + [02] α | 0 y
2 + [03] α | 0 y
3 + [04] α | 0 y
4 + · · ·)+
+
1
2!
∂
2 [µν] α
∂B 2
˛
˛
˛
0
( [01] α | 0 y + [02] α | 0 y
2 + [03] α | 0 y
3 + [04] α | 0 y
4 + · · ·)
2
+ · · · +
+
1
n!
∂
n [µν] α
∂B n
˛
˛
˛
0
( [01] α | 0 y + [02] α | 0 y
2 + [03] α | 0 y
3 + [04] α | 0 y
4 + · · ·)
n .
(20.121)