HARMONIC ANALYSIS AND PREDICTION OF TIDES 11 
34. The line PQ, joining P and Q, will then represent the direction 
and magnitude of the resultant force that tends to disturb the posi- 
tion of P relative to O, for it represents the difference between the 
force PC and a force through P equal and parallel to the force QC 
which acts upon 0. This last statement may be a little clearer to 
the reader if he will consider the force PC as being resolved into a 
force PD equal and parallel to QC, and the force PQ. The force 
PD, acting upon the particle at P, being equal and parallel to the 
force QC, acting upon a particle at O, will have no tendency to change 
the position of P relative to O. The remaining force PQ will tend 
to alter the position of P relative to O and is the tide-producing 
force of the moon at P. The force PQ may be resolved into a vertical 
component PR, which tends to raise the water at P, and the hori- 
zontal component PT’, which tends to move the water horizontally. 
FIGURE 2. 
35. If the point P’ is taken so that the distance CP’ is greater than 
the distance CO, the tide-producing force P’Q’ will be directed away 
from the moon. While at first sight this may appear paradoxical, 
it will be noted that the moon tends to separate O from P’, but as 
O is taken as the point of reference, this resulting force that tends to 
separate the points is considered as being applied at the point P’ 
only. 
36. To express the tide-producing force by mathematica] equations, 
refer to figure 2 and let 
r= OP =distance of particle P from center of earth, 
b= PC =distance of particle P from center of moon, 
d= OC =distance from center of earth to center of moon, 
2=COP=angle at center of earth between OP and OC. 
Also let 
M=mass of moon, 
E=mass of earth, 
a=mean radius of earth, 
u=attraction of gravitation between unit masses at unit 
distance. 
g=mean acceleration of gravity on earth’s surface. 
Since the force of gravitation varies directly as the mass and inversely 
as the square of the distance, 
Attraction of moon for unit mass at point O in direction oc" (2) 
