Geoscience Reference
In-Depth Information
which can be read from Fig. 1.10, considering that
√
u
2
+
E
2
is the semi-
major axis of the ellipsoid whose surface passes through
P
. Because of
ϑ
=90
◦
−
β
, we may equivalently write
x
=
√
u
2
+
E
2
cos
β
cos
λ,
y
=
√
u
2
+
E
2
cos
β
sin
λ,
(1-151)
z
=
u
sin
β.
Taking
u
= constant, we find
x
2
+
y
2
u
2
+
E
2
+
z
2
=1
,
(1-152)
u
2
which represents an ellipsoid of revolution. For
ϑ
= constant, we obtain
x
2
+
y
2
E
2
sin
2
ϑ
−
z
2
E
2
cos
2
ϑ
=1
,
(1-153)
which represents a hyperboloid of one sheet, and for
λ
= constant, we get
the meridian plane
y
=
x
tan
λ.
(1-154)
The constant focal length
E
, i.e., the distance between the coordinate origin
O
and one of the focal points
F
1
or
F
2
, which is the same for
all
ellipsoids
u
= constant, characterizes the coordinate system. For
E
=0wehavethe
usual spherical coordinates
u
=
r
and
ϑ, λ
as a limiting case.
To find
ds
, the element of arc, in ellipsoidal-harmonic coordinates, we
proceed in the same way as in spherical coordinates, Eq. (1-30), and obtain
ds
2
=
u
2
+
E
2
cos
2
ϑ
u
2
+
E
2
du
2
+(
u
2
+
E
2
cos
2
ϑ
)
dϑ
2
+(
u
2
+
E
2
)sin
2
ϑdλ
2
.
(1-155)
The coordinate system
u, ϑ, λ
is again orthogonal: the products
du dϑ
,etc.,
are missing in the equation above. Setting
u
=
q
1
,ϑ
=
q
2
,λ
=
q
3
,wehave
in (1-31)
h
1
=
u
2
+
E
2
cos
2
ϑ
2
=
u
2
+
E
2
cos
2
ϑ,
h
3
=(
u
2
+
E
2
)sin
2
ϑ.
(1-156)
,
u
2
+
E
2
If we substitute these relations into (1-32), we obtain
∂
∂u
(
u
2
+
E
2
)sin
ϑ
∂V
∂u
+
1
(
u
2
+
E
2
cos
2
ϑ
)sin
ϑ
∆
V
=
(1-157)
sin
ϑ
∂V
∂ϑ
+
u
2
+
E
2
cos
2
ϑ
(
u
2
+
E
2
)sin
ϑ
.
∂
∂ϑ
∂
∂λ
∂V
∂λ
