DEVELOPMENT OF TIDE-PRODUCING FORCE 
FUNDAMENTAL FORMULAS 
30. The tide-producing forces exerted by the moon and sun are 
similar in their action and mathematical expressions obtained for one 
may therefore by proper substitutions be adapted to the other. Be- 
cause of the greater importance of the moon in its tide-producing 
effects, the following development will apply primarily to that body, 
the necessary changes to represent the solar tides being afterwards 
indicated. 
31. The tide-producing force of the moon is that portion of its 
gravitational attraction which is effective in changing the water level 
on the earth’s surface. This effective force is the difference between 
the attraction for the earth as a whole and the attraction for the differ- 
ent particles which constitute the yielding part of the earth’s sur- 
face; or, if the entire earth were considered to be a plastic mass, the 
tide-producing force at any point within the mass would be the force 
that tended to change the position of a particle at that point relative 
to a particle at the center of the earth. That part of the earth’s 
surface which is directly under the moon is nearer to that body than 
is the center of the earth and is therefore more strongly attracted 
since the force of gravity varies inversely as the square of the dis- 
tance. For the same reason the center of the earth is more strongly 
attracted by the moon than is that part of the earth’s surface which 
is turned away from the moon. 
32. The tide-producing force, being the difference between the 
attraction for particles situated relatively near together, is small com- 
pared with the attraction itself. It may be interesting to note that, 
although the sun’s attraction on the earth is nearly 200 times as great 
as that of the moon, its tide-producing force is less than one-half 
that of the moon. If the forces acting upon each particle of the 
earth were equal and parallel, no matter how great those forces 
might be, there would be no tendency to change the relative posi- 
tions of those particles, and consequently there would be no tide- 
producing force. 
33. The tide-producing force may be graphically represented as in 
figure 2. 
Let O=the center of the earth, 
C=the center of the moon, 
P=any point within or on the surface of the earth. 
Then OC will represent the direction of the attractive force of the 
moon upon a particle at the center of the earth and PC the direction 
of the attractive force of the moon upon a particle at P. Now, let 
the magnitude of the moon’s attraction at P be represented by the 
length of the line PC. Then, since the attraction of gravitation varies 
inversely as the square of the distance, it is necessary, in order to 
represent the attraction at O on the same scale, to take a line CQ of 
such length that CQ : CP=CP? : CO’. 
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