Geology Reference
In-Depth Information
t
Z
P
D
r
Q
−
r
E
D
O
r
0
R
Figure 5.2 The integration point
Q
is a distance
D
from the field point
P
.
PQ
makes an angle
t
with the fixed line
OP
. The length of
PR
is
E
.
Then, using
OZ
as the axis of a system of spherical polar co-ordinates (
D
,
t
,
p
) with
P
as origin,
2π
π
E
D
2
sin
t
D
1
D
d
V=
dDdt dp
V
o
0
0
0
π
r
2
sin
2
t
2
r
0
−
=
π
−
r
cos
t
+
sin
tdt
0
r
2
π
0
2
r
π
0
cos
t
r
0
−
cos
2
t
sin
tdt
r
2
sin
2
t
sin
tdt
=
π
−
π
r
2
sin
2
t
sin
tdt
r
0
−
+
0
r
0
−
3
r
2
1
=
2π
.
(5.136)
Collecting the results of the foregoing integral transformations, we can write
(5.3) in the form
4π
G
2
2
r
0
−
3
r
2
P
2
(cosθ)
D
d
1
D
d
D
dS
1
V+
2
Ω
V+
+
4π
Ω
V−V
i
V
i
−V
o
S
i
⎩
−
4π
U
(
r
),
r
outside
S
i
,
=
(5.137)
−
4π
U
i
,
r
inside
S
i
,
after substitution for
W
(
r
) from equation (5.2). The advantage of this expression
is that none of the integrations include the region
r
<
c
inside the sphere with
radius equal to the polar radius of the equipotential
S
i
. Taking
r
<
c
, the latter of
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