Geology Reference
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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
d ¯ρ
dr 0
r 0
d ¯ρ
dr 0 .
ρ/¯ρ =
1
+
and
1
ρ/¯ρ =−
(5.175)
 
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