Geoscience Reference
In-Depth Information
familiar with the substitution
t
=cos
ϑ
and the corresponding equation for
g
(
t
).
Note that
f
(
τ
) satisfies formally the same differential equation as
g
(
t
),
namely, Legendre's equation (1-56). As we have seen, this differential equa-
tion has two solutions: Legendre's function
P
nm
and Legendre's function of
the second kind
Q
nm
.For
g
(
t
), where
t
=cos
ϑ
,the
Q
nm
(
t
) are ruled out for
obvious reasons, as we have seen in Sect. 1.8. For
f
(
τ
), however, both sets of
functions
P
nm
(
τ
)and
Q
nm
(
τ
) are possible solutions; they correspond to the
two different solutions
f
=
r
n
and
f
=
r
−
(
n
+1)
in the spherical case. Finally,
(1-169) has as before the solutions cos
mλ
and sin
mλ
.
We summarize all individual solutions:
f
(
u
)=
P
nm
i
u
E
or
Q
nm
i
u
E
;
(1-172)
g
(
ϑ
)=
P
nm
(cos
ϑ
);
h
(
λ
)=cos
mλ
or
sin
mλ .
Here
n
and
m<n
are integers 0
,
1
,
2
,...
, as before. Hence, the functions
V
(
u, ϑ, λ
)=
P
nm
i
u
E
P
nm
(cos
ϑ
)
cos
mλ
sin
mλ
,
(1-173)
P
nm
(cos
ϑ
)
cos
mλ
V
(
u, ϑ, λ
)=
Q
nm
i
u
E
sin
mλ
are solutions of Laplace's equation ∆
V
= 0, that is, harmonic functions.
From these functions we may form by linear combination the series
P
nm
i
u
E
V
i
(
u, ϑ, λ
)=
∞
n
i
·
b
E
P
nm
n
=0
m
=0
[
a
nm
P
nm
(cos
ϑ
)cos
mλ
+
b
nm
P
nm
(cos
ϑ
)sin
mλ
];
(1-174)
Q
nm
i
u
E
V
e
(
u, ϑ, λ
)=
∞
n
i
·
b
E
Q
nm
n
=0
m
=0
[
a
nm
P
nm
(cos
ϑ
)cos
mλ
+
b
nm
P
nm
(cos
ϑ
)sin
mλ
]
.
Here
b
is the semiminor axis of an arbitrary but fixed ellipsoid which may
be called the
reference ellipsoid
(Fig. 1.11). The division by
P
nm
(
ib/E
)or
Q
nm
(
ib/E
) is possible because they are constants; its purpose is to simplify
the expressions and to make the coecients
a
nm
and
b
nm
real.
