320 15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
P 0 (l, b) := x 0 ,
P 1 (l, b) := x 10 l + x 01 b =
α+β=1
x αβ l
α b
β ,
P 2 (l, b) := x 20 l
2 + x 11 lb + x 02 b
2 =
α+β=2
x αβ l
α b
β ,
. . .
P n (l, b) :=
α+β=n
x αβ l
α b
β ,
(15.31)
and
Q 0 (l, b) := y 0 ,
Q 1 (l, b) := y 10 l + y 01 b =
α+β=1
y αβ l
α b
β ,
Q 2 (l, b) := y 20 l
2 + y 11 lb + y 02 b
2 =
α+β=2
y αβ l
α b
β ,
. . .
Q n (l, b) :=
α+β=n
y αβ l
α b
β ,
(15.32)
subject to the Taylor expansion
r :=
G 11 /G 22 = cos B
1 − E
2 sin
2 B
1 − E 2
= r 0 + r 1 b + r 2 b
2 + r 3 b
3 + O(4) ,
(15.33)
namely
r 0 :=
1
0! r
(0) (B 0 ) = r(B 0 ) ,
r 1 :=
1
1! r
(1) (B 0 ) = r
(B 0 ) ,
. . .
r n :=
1
n! r
(n) (B 0 ) =
1
n(n−1)· ··· ·2·1 r
(n) (B 0 ) ,
(15.34)
and vice versa
s :=
G 22 /G 11 =
1
cos B
1 − E
2
1 − E 2 sin
2 B
= s 0 + s 1 b + s 2 b
2 + s 3 b
3 + O(4) =
= r
−1
0 − r
−2
0 r 1 b +
r
−3
0 r
2
1 − r
−2
0 r 2
b
2 +
−r
−4
0 r
3
1 + 2r
−3
0 r 1 r 2 − r
−2
0 r 3
b
2 + O(4) ,
(15.35)
namely
s 0 :=
1
0! s
(0) (B 0 ) = s(B 0 ) ,
s 1 :=
1
1! s
(1) (B 0 ) = s
(B 0 ) ,
. . .
s n :=
1
n! s
(n) (B 0 ) =
1
n(n−1)· ··· ·2·1 s
(n) (B 0 ) ,
(15.36)
given in detail by the coefficients of Box 15.1.