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Gm
l 2
x − ξ
l
Gm x − ξ
l 3
X =
F cos α =
=
,
Gm
l 2
y
η
Gm y
η
(1-4)
Y =
F cos β =
=
,
l
l 3
Gm
l 2
z
ζ
Gm z
ζ
Z =
F cos γ =
=
,
l
l 3
where
l = ( x − ξ ) 2 +( y − η ) 2 +( z − ζ ) 2 .
(1-5)
We next introduce a scalar function
V = Gm
l
,
(1-6)
called the potential of gravitation. The components X, Y, Z of the gravita-
tional force F are then given by
X = ∂V
∂x ,Y = ∂V
,Z = ∂V
∂z
,
(1-7)
∂y
as can be easily verified by differentiating (1-6), since
1
l
=
∂x
1
l 2
∂l
∂x =
1
l 2
x − ξ
l
x − ξ
l 3
=
,... .
(1-8)
In vector notation, Eq. (1-7) is written
F =[ X, Y, Z ] = grad V ;
(1-9)
that is, the force vector is the gradient vector of the scalar function V .
It is of basic importance that according to (1-7) the three components
of the vector F can be replaced by a single function V . Especially when we
consider the attraction of systems of point masses or of solid bodies, as we
do in geodesy, it is much easier to deal with the potential than with the three
components of the force. Even in these complicated cases the relations (1-7)
are applied; the function V is then simply the sum of the contributions of
the respective particles.
Thus, if we have a system of several point masses m 1 ,m 2 ,...,m n ,the
potential of the system is the sum of the individual contributions (1-6):
n
V = Gm 1
l 1
+ Gm 2
l 2
+ Gm n
l n
m i
l i
+
···
= G
.
(1-10)
i =1
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