Geoscience Reference
In-Depth Information
The left-hand side depends only on
u
and
ϑ
, the right-hand side only on
λ
.
The two sides cannot be identically equal unless both are equal to the same
constant. Therefore,
h
h
m
2
.
=
−
(1-164)
The factor by which
h
/h
is to be multiplied, i.e., the inverse of the main
factor on the left-hand side of (1-163), can be decomposed as follows:
u
2
+
E
2
cos
2
ϑ
(
u
2
+
E
2
)sin
2
ϑ
=
E
2
u
2
+
E
2
.
1
sin
2
ϑ
−
(1-165)
Substituting (1-164) and (1-165) into (1-163) and combining functions of
the same variable, we obtain
E
2
m
2
sin
2
ϑ
.
(1-166)
1
1
f
[(
u
2
+
E
2
)
f
+2
uf
]+
u
2
+
E
2
m
2
=
g
(
g
+
g
cot
ϑ
)+
−
The two sides are functions of different independent variables and must there-
fore be constant. Denoting this constant by
n
(
n
+ 1), we finally get
u
2
+
E
2
m
2
f
(
u
) = 0 ; (1-167)
sin
ϑg
(
ϑ
)+cos
ϑg
(
ϑ
)+
n
(
n
+1)sin
ϑ
n
(
n
+1)
E
2
(
u
2
+
E
2
)
f
(
u
)+2
uf
(
u
)
−
−
g
(
ϑ
) = 0 ;
m
2
sin
ϑ
−
(1-168)
h
(
λ
)+
m
2
h
(
λ
)=0
.
(1-169)
These are the three ordinary differential equations into which the partial
differential equation (1-159) is decomposed by the separation of variables
(1-161).
The second and third equations are the same as in the spherical case,
Eqs. (1-46) and (1-47); the first equation is different. The substitutions
τ
=
i
u
E
(where
i
=
√
−
1) and
t
=cos
ϑ
(1-170)
transform the first and second equations into
f
(
τ
)+
n
(
n
+1)
τ
2
f
(
τ
)=0
,
m
2
τ
2
)
f
(
τ
)
(1
−
−
2
τ
−
1
−
(1-171)
(1
− t
2
)
g
(
t
)
−
2
t g
(
t
)+
n
(
n
+1)
−
t
2
g
(
t
)=0
,
m
2
1
−
where the overbar indicates that the functions
f
and
g
are expressed in terms
of the new arguments
τ
and
t
. From spherical harmonics we are already
