Gravity field of the earth - Physical Geodesy - page 92

∂T

∂h + 1

∂γ
∂h T,

∆ g = − (2-248)

γ

∂γ
∂h N,

δg =∆ g − (2-249)

1
γ

∂γ
∂h T,

δg =∆ g − (2-250)

relating different quantities of the anomalous gravity field.
Another equivalent form is
∂T
∂h −

1
γ

∂γ
∂h T +∆ g =0 .

(2-251)

This expression has been called the fundamental equation of physical geodesy ,
because it relates the measured quantity ∆ g to the unknown anomalous
potential T . In future, the relation
∂T
∂h + δg = 0

(2-252)

may replace it.
It has the form of a partial differential equation. If ∆ g were known
throughout space, then (2-251) could be discussed and solved as a real par-
tial differential equation. However, since ∆ g is known only along a surface
(the geoid), the fundamental equation (2-251) can be used only as a bound-
ary condition , which alone is not sucient for computing T . Therefore, the
name “differential equation of physical geodesy”, which is sometimes used
for (2-251), is rather misleading.
One usually assumes that there are no masses outside the geoid. This is
not really true. But neither do we make observations directly on the geoid;
we make them on the physical surface of the earth. In reducing the measured
gravity to the geoid, the effect of the masses outside the geoid is removed by
computation, so that we can indeed assume that all masses are enclosed by
the geoid (see Chaps. 3 and 8).
In this case, since the density is zero everywhere outside the geoid, the
anomalous potential T is harmonic there and satisfies Laplace's equation

∂ 2 T
∂x 2

+ ∂ 2 T
∂y 2

+ ∂ 2 T
∂z 2

∆ T ≡ =0 . (2-253)

This is a true partial differential equation and su ces, if supplemented by
the boundary condition (2-251), for determining T at every point outside
the geoid. If we write the boundary condition in the form

∂T

∂n + 1

∂γ
∂n T =∆ g,

− (2-254)

γ