The pure gravitational attraction between two small bodies of masses m and m' separated by a distance r is given by the Newtonian law of attraction
where F is the gravitational force and G is the universal gravitational constant
The force is directed along the straight line connecting the centers of mass of the two bodies.
Eq. (1) is applicable to bodies whose dimensions are extremely
small compared with the distance r. It can be shown, however,
that the attraction 
of a unit mass which is external to a
sphere of uniform density is
where M is the total mass of the sphere and r is the distance of the unit mass from the center of the sphere. This equation is applicable if the sphere is radially stratified (i.e., its density depends only on the distance from the center(.
If the earth is considered as a radically symmetric sphere as a first approximation, then a unit mass at sea level is subject to the absolute gravitational attraction
where a is the mean radius of the earth and 
is the
mass of the earth. The values of a and are included in
Table 2.01-1, along with some other important physical
characteristics of the earth.
Figure 2.01-1a,b,c illustrates the three cases discussed above.
As can be seen from Table 2.01-1, the mass of the atmosphere is
less than one-millionth part of the earth as a whole and its
effect can be neglected in determining the variation of
with elevation above sea level. If we let z represent
elevation above sea level then a unit mass located at
altitude z will be at a distance
from the earth's center, where a may be taken as the mean radius of the earth as in the approximation (3). Therefore, from Eq. (2) and (3) we find that at altitude z above the earth's surface
For small values of z compared with a, the above equation can be
simplified by the binomial theorem. Neglectting terms in
and higher order we have
This applies in the atmosphere only. The value of 2/a is
m 
. Consequently, even at
an altitude of 10,000 meters (about 33,000 feet) where
the atmospheric pressure is only about one-quarter that at the
surface, the absolute gravitational attraction is decreased by only
0.3 of one per cent below that at sea level (this amounts to
a reduction of 3 dynes/gm below the mean value of 982.2 dynes/gm
at sea level). Indeed, a layer of air of 10,000 meters thickness
barely contains the highest topographic anomaly of the earth
(Mt. Everest).
The variation of with depth in the ocean or in the lithosphere
cannot be computed from Eq. (6) because of the influence of the mass
of the material above the depth in question. The approximate
radial distribution of density in the earth is given
in Table 2.01-II. This muts be taken into account in
attempting to compute the value of at different depths.
If we let m represent the mass of the nearly spherical shell of the earth above a depth -z (referred to mean sea level) it can be shown that
or, making use of Eq. (3)
Equation (7) applies strictly to a radially stratified spherical mass and therefore definitely constitutes an approximation for the earth.
We can write
where 
is the mean density of the
earth as a whole. For small |z|
where 
is the mean density of the outer spherical
shell of the earth of thickness z. If we introduce these
relations in Eq. (8) and simplify (neglecting
squares of z/a and higher orders) we obtain
In general we can write
for all z where
Table 2.01-A gives approximate values of 
in the air, sea and
the earth's crust.
The values of and
used in ascertaning itme (iii) of the
above table are 2.7 and 5.52 gm/cm 
, respectively. In item
(ii) the value of is taken as 1.04 gm/cm 
.
The above discussion deals with the variation of in a small
layer in the immediate vicinity of the earth's surface. It is of
interest to see how this fits
into the broad scale variation of . Obviously at the center
of the earth there can be no net attractive force, and since the
value of increases with depth near the surface there must
exist a maximum of inside the earth itself.
If we let now represent the mean density of the
earth's material contained within a sphere of radius r,
then it can be shown that the maximum value of occurs at
that r for which
and the corresponding maximum value of is
A plot of the density 
and the mean density
are shown in Fig. 2.01-2. The associated
distribution of is shown in the same figure.
