If we apply the conditions required for static equlibrium of the ocean in the presence of the tide producing forces at a given instant, it is possible to evaluate the equilibrium form of the surface of an ocean covering the entire earth. The deviation of this imaginary equilibrium form under the influence of the frozen tide producing forces from the equlibrium form under the influence of gravity alone represents, in effect, the potential of the tide producing force.
If we apply the conditions required for static equilibrium of the ocean in the presence of the tide-producing forces at any given instant, it is possible to evaluate the equilibrium form of the surface of an ocean covering the entire earth. The deviation of this imaginary equilibrium form under the influence of the ``frozen'' tide-producing forces from the equilibrium form under the influence of gravity alone represents in effect the potential of the tide-producing forces.
Let 
represent the vertical displacement of sea level from
the level surface which is defined by gravity alone. Under
conditions of static equilibrium, the sea surface must be normal
to the effective gravity which includes the tide-producing
force.
From Fig. 2.04-6 is is evidence that at a given position of the surface
The distance x is to be measured along a great circle path which lies in the plane which, if extended, passes through the center of the body producing the tidal force (Fig. 2.04-4a). Thus, if we take the reference surface as a sphere then
Consequently, the potential associated with the lunar
tide-producing force (Eq. 50b) is governed by the equation
the negative sign being required since the force is directed
opposite to the direction of increasing 
. Integrating
this gives
The constant of integration must be chosen such that no change in the volume of water results from the tidal distortion. This requires that C be unity such that
It is interesting to note that this has the same form as the vertical component of the tidal force (Eq. 50a). In fact,
even though 
has nothing to do with actually producing
the distortion. The maximum value of is 35.4 cm and occurs
at values of equal to 0 and 
(i.e. directly under
the moon and opposite). The minimum value of occurs for
equal to 
or 
and equals -17.7 cm.
Thus, the range of the tidal distortion is 53.1 cm. The form of
the tidal equilibrium surface is a prolate spheriod of
revolution whose axis of symmetry is always directed towards
the moon. This is illustrated in Fig. 2.04-7. The value
of is zero for 
.
The potential or equilibrium distortion associated with the solar tide-producing forces is given by
The range of 
is 24.4 cm. At the times of the solar
and lunar eclipses the total potential distortion due to the
combined action of the sun and the moon acquires the maximum
range of 77.5 cm (2.54 feet). At the times of new moon and
full moon, which occur about every 14 days, the combined tidal potential
approaches this extreme condition but the effect is generally somewhat
less because of the inclination of the moon's orbit to the earth/sun
ecliptic plane.
For a perspective diagram of the distribution of the horizontal tide-producing forces reference is made to (), Fig. 138B, p. 547.