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while equation (3.36) is reduced to
∂v
2μ)
∂
u
n
∂
r
n
∂
u
n
∂
r
∂y
4
∂
r
+
1
r
y
2
+
1
r
y
4
+
2
μ
r
λ
2
r
y
2
−
(λ
+
+
∂
r
−
1
1
r
ρ
0
φ
r
ρ
0
g
0
u
n
−
n
Fn
,
=
−
v
(3.49)
and equation (3.37) is reduced to
∂
z
2
r
2
n
(
n
1
t
n
+
3
r
∂
t
n
t
Fn
.
∂
r
−
+
1
)
+
∂
r
=−
(3.50)
Recast in this form, the radial spheroidal, transverse spheroidal and torsional
parts of the equilibrium equation are free of derivatives of the Earth model proper-
tiesλ,μ,ρ
0
andg
0
. From the numerical point of view, the advantages of this form of
the equations were recognised by Alterman, Jarosch and Pekeris (Alterman
et al
.,
1959) and led to their
y-notation
in which y
1
u
n
denotes the radial coe
=
cient
n
denotes the radial coe
of the radial spheroidal displacement, y
3
=
v
cient of the
n
denotes the radial coe
transverse spheroidal displacement andy
5
cient of the
decrease in gravitational potential. The six variables y
1
, y
2
, y
3
, y
4
, y
5
, y
6
describe
the spheroidal deformation field. From equation (3.41), we obtain
∂y
1
∂
r
=−
=
φ
2λ
(λ
+
1
λ
+
n
(
n
+
1)λ
2μ)
r
y
1
+
2μ
y
2
+
2μ)
r
y
3
.
(3.51)
(λ
+
Using this equation and (3.45), we can rewrite equation (3.48) as
2μ)
r
2
2g
0
+
r
Ω
2
∂y
2
∂
r
=
2
ρ
0
r
(3λ
+
2μ)
4μ
(λ
+
−
+
4μ
y
1
−
2μ)
r
y
2
(λ
+
n
(
n
+
1)
r
ρ
0
g
0
2
n
(
n
+
1)
(3λ
+
2μ)
n
(
n
+
1)
r
y
4
+
−
μ
y
3
+
r
2
(λ
+
2μ)
u
Fn
.
−
ρ
0
y
6
−
(3.52)
Equation (3.42) gives
∂y
3
∂
r
=−
1
r
y
1
+
1
r
y
3
+
1
μ
y
4
.
(3.53)
Using this equation and (3.51), we can rewrite equation (3.49) as
ρ
0
g
0
r
−
2μ)
r
2
∂y
4
∂
r
=
(3λ
+
2μ)
λ
(λ
+
2μ
y
1
−
2μ)
r
y
2
(λ
+
2μ)
r
2
2
n
2
1
λ
+
2
n
2
1
μ
y
3
2μ
(λ
+
+
+
2
n
−
+
n
−
3
r
y
4
−
ρ
0
m
−
r
y
5
−
v
Fn
.
(3.54)
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