Geography Reference
In-Depth Information
n
=4:
12
s
0
x
40
+2
r
0
x
22
+
r
1
x
21
=0
,
(15.58)
12
s
0
y
40
+2
r
0
y
22
+
r
1
y
21
=0
,
(15.59)
3
s
0
x
31
+3
s
1
x
30
+3
r
0
x
13
+2
r
1
x
12
+
r
2
x
11
=0
,
(15.60)
3
s
0
y
31
+3
s
1
y
30
+3
r
0
y
13
+2
r
1
y
12
+
r
2
y
11
=0
,
(15.61)
2
s
0
x
22
+2
s
1
x
21
+2
s
2
x
20
+12
r
0
x
04
+9
r
1
x
03
+6
r
2
x
02
+3
r
3
x
01
=0
,
(15.62)
2
s
0
y
22
+2
s
1
y
21
+2
s
2
y
20
+12
r
0
y
04
+9
r
1
y
03
+6
r
2
y
02
+3
r
3
y
01
= 0;
(15.63)
and in general
∞
n−
2
i
sx
ll
+(
rx
b
)
b
=
[(
j
+1)[(
i
−
j
+1)
r
j
+1
x
n−i−
2
,i−j
+1
+
n
=2
i
=0
j
=0
1)
s
i
x
n−i,i−j
]
l
n−i−
2
b
i
=0
,
+(
j
+2)
r
i−j
x
n−i−
2
,j
+2
]+(
n
−
i
)(
n
−
i
−
(15.64)
∞
n−
2
i
sy
ll
+(
ry
b
)
b
=
[(
j
+1)[(
i
−
j
+1)
r
j
+1
y
n−i−
2
,i−j
+1
+
n
=2
i
=0
j
=0
1)
s
i
y
n−i,i−j
]
l
n−i−
2
b
i
=0
.
+(
j
+2)
r
i−j
y
n−i−
2
,j
+2
]+(
n
−
i
)(
n
−
i
−
End of Corollary.
Finally, we here have to constrain the general solution
x
(
l,b
) of the Laplace-Beltr
ami equ
a-
tion (compare with (
15.29
)) to the ellipsoidal Korn-Lichtenstein equation
y
l
=
−
G
11
/G
22
x
b
(compare with (
15.27
)), in particular
y
l
=
−r
(
b
)
x
b
=
−
(
r
0
+
r
1
b
+
r
2
b
2
+
r
3
b
3
+ O(4))
x
b
,
(15.65)
y
l
=
y
10
+2
y
20
l
+
y
11
b
+3
y
30
l
2
+2
y
21
lb
+
y
12
b
2
+4
y
40
l
3
+3
y
31
l
2
b
+2
y
22
lb
2
+
y
13
b
3
+O(4)=
r
0
x
21
l
2
=
−
r
0
x
01
−
r
o
x
11
l
−
2
r
0
x
02
b
−
r
1
x
01
b
−
−
2
r
0
x
12
lb
−
r
1
x
11
lb
−
−
3
r
0
x
03
b
2
−
2
r
1
x
02
b
2
− r
2
x
01
b
2
− r
0
x
31
l
3
−
2
r
0
x
22
l
2
b − r
1
x
21
l
2
b −
3
r
0
x
13
lb
2
−
(15.66)
2
r
1
x
12
lb
2
r
2
x
11
lb
2
4
r
0
x
o
4
b
3
3
r
1
x
03
b
3
2
r
2
x
02
b
3
r
3
x
01
b
3
+O(4)
,
−
−
−
−
−
−
Alternatively, w
e here have
to constrain the general solution to the ellipsoidal Korn-Lichtenstein
equation
y
b
=
−
G
22
/G
11
x
l
(compare with (
15.27
)), in particular
y
b
=
s
(
b
)
x
l
=(
s
0
+
s
1
b
+
s
2
b
2
+
|s
3
b
3
+ O(4))
x
l
,
(15.67)
y
b
=
y
01
+
y
11
l
+2
y
02
b
+
y
21
l
2
+2
y
12
lb
+3
y
03
b
2
+
y
31
l
3
+2
y
22
l
2
b
+3
y
13
lb
2
+4
y
04
b
3
+O(4)=
=
s
0
x
10
+2
so
x
20
l
+
s
0
x
11
b
+
s
1
x
10
b
+3
s
0
x
30
l
2
+2
s
0
x
21
lb
+
s
1
x
20
lb
+
+
s
0
x
12
b
2
+
s
1
x
11
b
2
+
s
2
x
10
b
2
+4
s
0
x
40
l
3
+3
s
0
x
31
l
2
b
+3
s
1
x
30
l
2
b
+2
s
0
x
22
lb
2
+
(15.68)
+2
s
1
x
21
lb
2
+2
s
2
x
20
lb
2
+
s
0
x
13
b
3
+
s
1
x
12
b
3
+
s
2
x
11
b
3
+
s
3
x
10
b
3
+O(4)
.
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