30 U. S. COAST AND GEODETIC SURVEY 
of nature does not even approximate to that which might be expected 
under the assumed conditions, the theory is of value as an aid in 
visualizing the distribution of the tidal forces over the surface of the 
earth. The theoretical tide formed under these conditions is known 
as the equilibrium tide, and sometimes as the astronomical or gravita- 
tional tide. 
89. Under the equilibrium theory, the moon would tend to draw 
the earth into the shape of a prolate spheroid with the longest axis 
in line with the moon, thus producing one high water directly under 
the moon and another one on the opposite side of the earth with a 
low water belt extending entirely around the earth in a great circle 
midway between the high water points. It may be shown mathe- 
matically, however, that the total effect of the moon at its mean dis- 
tance would be to raise the high water points about 14 inches above 
the mean surface of the earth and depress the low water belt about 
7 inches below this surface, giving a maximum range of tide of about 
21 inches. The corresponding range due to the sun is about 10 inches. 
Figures 5 and 6 illustrate on an exaggerated scale the theoretical 
disturbing effect of the moon on the earth. In the first figure the 
moon is assumed to be directly over the equator and in the last figure 
the moon is approximately at its greatest north declination. 
90. With the moon over the equator (fig. 5), the range of the equi- 
librium tide will be at a maximum at the equator and diminish to 
zero at the poles and at any point there wil] be two high and low 
waters of equal range with each rotation of the earth. With the 
moon north or south of the equator (fig. 6), a declinational inequality 
is introduced and the two high and low waters of the day for any given 
latitude would no longer be equal except at the equator. This 
inequality would increase with the latitude and near the poles only 
one high and low water would occur with each rotation of the earth. 
Although latitude is an important factor in determining the range of 
the equilibrium tide, it is to be kept in mind that in the actual tide 
of nature the latitude of a place has no direct effect upon the rise and 
fall of the water. 
91. A surface of equilibrium is a surface at every point of which the 
sum of the potentials of all the forces is a constant. On such a 
surface the resultant of all the forces at each point must be in the 
direction of the normal to the surface at that point. If the earth 
were a homogeneous mass with gravity as the only force acting, the 
surface of equilibrium would be that of a sphere. Each additional 
force will tend to disturb this spherical surface, and the total deforma- 
tion will be represented by the sum of the disturbances of each of the 
forces acting separately. In the following investigation we need not 
be especially concerned with the more or less permanent deformation 
due to the centrifugal force of the earth’s rotation, since we may 
assume that the disturbances of this spheriodal surface due to the 
tidal forces will not differ materially from the disturbances in a true 
spherical surface due to the same cause. 
92. The potential at any point due to a force is the amount of work 
that would be required to move a unit of matter from that point, 
against the action of the force, to a position where the force is zero. 
This amount of work will be independent of the path along which 
the unit of matter is moved. If the force being considered is the 
gravity of the earth the potential at any point will be the amount 
