In order to investigate the broad scale configuration of the
equigeopotential surfaces, it is convenient to introduce two
additional potential functions. Gravity actually represents
the vector sum of the two forces 
and
(Art. 2.02). Each of those forces are
conservative and can be represented in terms of
potential functions. We will define 
and 
such that
and
Consequently,
or
The constant can be taken as zero with no loss of generality.
Eq. (69) is quite useful since it allows for a determination
of the field of 
by graphical addition of the
equiscalar surfaces of and , which are rather
simple geometric forms.
The evaluation of the absolute gravitational potential
can be carried out as follows. Considering the earth as a
sphere, is directed radially towards the
center at every point. Consequently, the scalar equivalent
of Eq. (66) is
for positions exterior to the earth where r is the radial
distance from the center of the earth.
Integrating and setting equal to zero at the
position 
gives
Thus the equiscalar surfaces of are concentric spheres
whose geometrical spacing increases with increasing distance
from the center. Solving for r yields
where 
is the dimensionless quantity
where 
is the mean value of gravity at sea level.
The potential of the centrifugal force associated
with a unit mass having an angular velocity about the earth's
axis equal to that of the earth is determined by
where R represents the radial distance from the earth's axis
to the unit mass measured normal to the earth's axis. The
negative sign in Eq. (74) is required in view of Eq. (67) because
is directed outwards. Thus must
decrease with increasing R since 
points towards low
values of 
.
Integrating Eq. (74) yields
where is arbitrarily taken as zero at the poles.
The equiscalar surfaces are concentric cylinders about
the earth's axis. The geometrical spacing decreases with
increasing distance from the axis. Solving for R obtains
where 
is a negative dimensionless quantity given by
A plot of isolines of and for a plane section
through the center of the earth is shown in Fig. 2.07-1.
The isolines of have been constructed by graphical addition
of the two component potentials.
The particular surface
represents the approximate equilibrium form
of the sea surface under perfectly hydrostatic conditions in the
absence of tidal forces, currents, etc. Taking 
where is the latitude and rearranging Eq. (78) gives
At 
, while at the equator
The second term on the right is very small compared with a since it is known that the level surfaces depart only slightly from a spherical shape. Consequently, for all practical purposes r can be taken as equal to a on the right, i.e.
at the equator. The magnitude of the second term on the
right is 10.75 km. Actually, the difference between the
mean equatorial and polar radii is about 21.5 km or just twice
the value above. The discrepancy is due to the fact that 
itself varies with latitude (see Art. 2.02).
The actual variation of r is therefore more accurately given by
