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solution, v
2
(
a
)
0every-
where, while y
3
and y
4
do not appear. The system (7.41) then degenerates to
=−
y
52
(
a
), a constant, and w
2
(
a
)
=
0. In both solutions, y
6
=
=
χ
0
(
a
+
),
D
1
v
1
(
a
)
+
D
2
v
2
(
a
)
dV
1
,
0
dr
(
a
+
).
D
1
w
1
(
a
)
=
(7.44)
Without loss of generality, the constant v
2
(
a
) can be absorbed into the linear com-
bination constant
D
2
, giving
D
2
=
χ
0
(
a
+
),
D
1
v
1
(
a
)
+
dV
1
,
0
dr
(
a
+
).
=
D
1
w
1
(
a
)
(7.45)
The systems (7.34), (7.36), (7.41) and (7.45) are all seen to depend only on the
boundary values of χ
n
and the derivatives of
V
1,
n
. Thus, the entire deformation field
in either the shell or the inner core is determined by the values of these two quantit-
ies on the boundaries. Following the traditional definition of the Love number
h
as
a dimensionless number giving the radial displacement as a proportion of the ratio
of the disturbing potential to gravity, we write the radial coe
cients of the degree-
n
parts of the respective radial displacement components on the two boundaries as
h
1
n
χ
n
(
b
−
)
(
b
−
)
1
g
0
(
b
)
dV
1
,
n
dr
bh
2
n
y
1,
n
(
b
)
=
−
(7.46)
and
h
1
n
χ
n
(
a
+
)
(
a
+
)
1
g
0
(
a
)
dV
1
,
n
dr
−
ah
2
I
y
1,
n
(
a
)
=
.
(7.47)
n
For each spherical harmonic of degree
n
, the Love numbers
h
1
n
,
h
2
n
for the shell,
and the Love numbers
h
1
n
,
h
2
I
n
for the inner core, are found by numerical integra-
tion of the sixth-order spheroidal system of di
ff
erential equations, (3.102) through
(3.107), for
n
≥
1, or the fourth-order spheroidal system, (3.110) through (3.113),
for
n
0. Only the conditions of continuity of normal displacement at the two
solid-fluid interfaces remain to be applied.
Application of the subseismic form (6.181) of the equation of continuity provides
an additional constraint. In terms of the radial coe
=
cients of the degree-
n
vector
spherical harmonics, it becomes
y
2
(
r
)
=
ρ
0
(
r
)g
0
(
r
)y
1
(
r
),
(7.48)
for
a
≤
r
≤
b
. At the core-mantle boundary this yields the constraint
1
ρ
0
(
b
−
)
C
2
g
0
(
b
)
C
1
−
=
0,
(7.49)
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