Geoscience Reference
In-Depth Information
The fact that
r
is used both as an integration variable and as an upper
limit should not cause any diculty. Substituting the upward continuation
integral (2-282), we get
⎡
⎤
cos
ψ
∆
gdσ
r
r
2
T
=
R
2
4
π
r
3
R
2
r
l
3
−
+1+
3
R
r
⎣
⎦
dr .
−
(2-296)
∞
σ
Interchanging the order of the integrations gives
r
2
T
=
R
2
4
π
r
cos
ψ
dr
∆
gdσ.
r
3
R
2
r
l
3
−
+1+
3
R
r
−
(2-297)
∞
σ
The integral in brackets can be evaluated by standard methods. The indefi-
nite integral is
cos
ψ
dr
r
3
R
2
r
l
3
−
+1+
3
R
r
−
2
r
2
l
R
cos
ψ
+
l
)+
r
+3
R
cos
ψ
ln
r.
(2-298)
The reader is advised to perform this integration, taking into account (2-
275), or at least to check the result by differentiating the right-hand side
with respect to
r
.
For large values of
r
,wehave
l
=
r
1
=
−
3
l
−
3
R
cos
ψ
ln(
r
−
=
r
R
r
−
cos
ψ
···
−
R
cos
ψ
···
(2-299)
and, hence, we find that as
r →∞
, the right-hand side of the above indefinite
integral approaches
5
R
cos
ψ −
3
R
cos
ψ
ln 2
.
(2-300)
If we subtract this from the indefinite integral, we get the definite integral,
since infinity is its lower limit of integration. Thus,
r
cos
ψ
dr
r
3
R
2
r
l
3
−
+1+
3
R
r
−
(2-301)
∞
R
cos
ψ
5+3ln
r
.
2
r
2
l
−
R
cos
ψ
+
l
2
r
=
+
r
−
3
l
−
Hence, we obtain Pizzetti's formula
T
(
r, ϑ, λ
)=
R
4
π
S
(
r, ψ
)∆
gdσ,
(2-302)
σ
