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H
KL
U
K
U
L
+
Q
H
NQ
U
K
U
L
U
N
KL
τ
g
=
G
M
1
M
2
(
M
1
+
U
M
2
)
⇒
(20
.
30)
.
+
U
M
1
)(
M
2
K
1
L
1
K
2
L
2
End of Proof.
E-2 The Differential Equations of Third Order of a Geodesic Circle
For the proof of Corollary
20.2
,wedepartfromthe
Darboux derivational equations
under the
postulate of
geodesic circle κ
g
=const.,
κ
n
= const., and
τ
g
=0,namely
D
1
=
(
κ
g
+
κ
n
)
D
1
=
(
κ
g
+
κ
n
)
U
M
G
M
.
−
−
(E.14)
Proof (Corollary
20.2
).
D
1
=
U
M
+3
U
K
U
L
M
+
U
K
U
L
U
N
KL
M
KL
− H
KL
H
NQ
G
QM
G
M
+
Q
M
NQ
KL
,N
=
⇒
(E.15)
(
κ
g
+
κ
n
)
U
M
=
U
M
+3
U
K
U
L
+
U
K
U
L
U
N
M
KL
H
KL
H
NQ
G
QM
,
+
Q
M
NQ
−
KL
,N
U
K
1
U
L
1
M
1
+
U
M
1
U
K
2
U
L
2
M
2
+
U
M
2
+
κ
g
+
κ
n
=
G
M
1
M
2
K
1
L
1
K
2
L
2
+
H
K
1
L
1
H
K
2
L
2
U
K
1
U
K
2
U
L
1
U
L
2
=
⇒
+
U
K
U
L
U
N
M
KL
U
M
+3
U
K
U
L
M
KL
+
Q
KL
M
NQ
U
K
U
L
U
N
H
KL
H
NQ
G
QM
+
−
,N
U
K
1
U
L
1
M
1
K
1
L
1
+
U
M
1
U
K
2
U
L
2
M
2
K
2
L
2
U
M
2
U
M
+
(E.16)
+
G
M
1
M
2
+
H
K
1
L
1
H
K
2
L
2
U
K
1
U
K
2
U
L
1
U
L
2
U
M
=0
⇐⇒
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