HARMONIC ANALYSIS AND PREDICTION OF TIDES 31 
of work required to move a unit mass against the force of gravity 
from the point to an infinite distance from the earth’s center. For 
the tide-producing force, the potential at any point will be measured 
by the amount of work necessary to move the unit of mass to the 
earth’s center where this force is zero. 
93. Referring to formula (21) for the vertical component of the tide- 
producing force, if the unit g is replaced by the unit » from equation 
(15), the formula may be written as follows: 
Sul 
a ese z—1/3)r+—— odF 
(5 cos*? z—3 cos 2)r? (109) 
94. Considering separately the tide-producing potential due to the 
two terms in the above formula, let the potential for the first term 
involving the cube of the moon’s distance be represented by V3 and 
the potential for the second term involving the 4th power of the 
moon’s distance by Vy. In each case the work required to move a unit 
mass against the force through an infinitesimal distance —dr toward 
the center of the earth is the product of the force by —dr, and the 
potential or total work required to move the particle to the center of 
the carth may be obtained by integrating between the limits rand 
zero. Thus 
‘== ee (cos? 21/3) |r dr 
=e (cos? z—1/3)r (110) 
Ve= ag cos’ z—3 cos 2) ie dr 
mG cos’ z—3 cos z)r’ (111) 
95. At any instant of time the tide-producing potential at different 
points on the earth’s surface will depend upon the zenith distance (2) 
of the moon and may be either positive or negative. It will now be 
shown that the average tide-producing potential for all points on the 
earth’s surface, assuming it to be a sphere, is zero. Assume a series 
‘of right conical surfaces with common apex at center of earth and axis 
coinciding with the line joining centers of earth and moon, the angle 
between the generating line and the axis being z. These conical 
surfaces separated by infinitesimal angle dz will cut the surface of the 
sphere into a series of equipotential rings, the surface area of any ring 
being equal toa 2 mr’ sin zdz. The average potential for the entire 
spherical surface may then be obtained by summing the products of 
the ring areas and corresponding potentials and dividing the sum by 
the total surface area of the sphere. Thus 
Average Vin 28 | "(cost z—1/3) sin z dz 
0 
pe T 
a | -13 cos’ z+1/3 cos z[=0 (112) 
