Geoscience Reference
In-Depth Information
P
0
l
l'
R
r
Q'
Ã
Q
()
dm
r'
(')
dm
Fig. 3.13. Rudzki reduction as an inversion in a sphere
these mass elements at the geoidal point
P
0
is
=
G
dm
l
Gdm
r
2
+
R
2
dU
T
=
2
Rr
cos
ψ
,
−
(3-91)
dU
C
=
G
dm
l
Gdm
r
2
+
R
2
=
cos
ψ
.
−
2
Rr
We should have
dU
C
=
dU
T
(3-92)
if
dm
=
R
r
dm
(3-93)
and
r
=
R
2
r
.
(3-94)
This is readily verified by substitution into the second equation of (3-91).
The condition (3-94) means that
Q
and
Q
are related by
inversion in the
sphere
of radius
R
(Kellogg 1929: p. 231). Therefore, this reduction method
is called
inversion reduction or Rudzki reduction
.
The condition (3-93) expresses the fact that the compensating mass
dm
is not exactly equal to
dm
but is slightly smaller. Since this relative decrease
of mass is of the order of 10
−
8
, it may be safely neglected by setting
dm
=
dm .
(3-95)
Usually it is even sucient to replace the sphere by a plane. Then
Q
is the
ordinary mirror image of
Q
(Fig. 3.14).
