Geology Reference
In-Depth Information
5
4
7
κ.
42
f
2
e
=
f
−
+
(5.168)
Then,
5
4
7
κ,
42
f
2
f
=
e
+
−
(5.169)
5
4
f
=
e
+
21
ff
−
7
κ
,
(5.170)
5
5
4
f
=
e
+
21
f
2
21
ff
−
7
κ
.
+
(5.171)
As before,
f
in the second-order term on the right side of (5.171) may be replaced
by its first-order approximation given by equation (5.165), and the derivatives of κ
in (5.171) and (5.170) can be replaced through expression (5.166). Further, since
e
di
ers from
f
only in second-order quantities, we can use it in place of
f
in
any second-order term and commit at most a third-order error. Substituting expres-
sions (5.169) through (5.171) into the second-order Clairaut equation (5.167) and
collecting terms, we find, correct to second order, that it becomes
ff
r
0
ρ
¯ρ
6
6
r
0
e
+
e
−
(1
−
ρ/¯ρ)
e
−
ρ/¯ρ
)
⎩
⎭
.
(5.172)
4
7
(
1
6
7
r
0
7
m
r
0
e
r
0
e
3
e
2
r
0
ee
+
e
2
=−
+
−
+
The e
ect of this change of variable is to absorb κ into the new variable
e
and to
make (1
ff
−
ρ/¯ρ) a common factor of the second-order terms. A further transform-
ation, due to Radau, is to make a variable change of the type used to linearise the
Riccati equation,
r
0
e
e
.
η
=
(5.173)
With this change of independent variable, we have
2
e
r
0
η and
e
r
0
η
+
η
r
0
−
r
0
e
=
e
=
.
(5.174)
Further, rearranging (5.162) we obtain
r
0
3ρ
d
¯ρ
dr
0
r
0
3ρ
d
¯ρ
dr
0
.
ρ/¯ρ
=
1
+
and
1
−
ρ/¯ρ
=−
(5.175)
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