PROGRESS OF OUR KNOWLEDGE OF THE TIDES. 123 
The tides of many portions of the ocean may be illustrated 
by a pendulum to which small horizontal forces are applied, 
whose periods are very nearly the free period of the pendu¬ 
lum. The result, as stated in Rayleigh’s Theory of Sound, 
will be that the phase of the vibration of the pendulum will 
be very nearly one-fourth of a period behind the phase of 
the forces; whereas the corrected equilibrium theory re¬ 
quires an agreement of phases, and the theory of forced 
oscillations requires opposition or agreement of phases, ac¬ 
cording to the dimensions of the body of water. 
For the purpose of illustration, suppose we have an east 
and west canal extending along the equator whose length 
is one-half the distance traveled by a free wave during 12 
lunar hours. The depth of the canal is not important, so 
long as it is not exceedingly shallow, for the length of the 
canal, its period being fixed, is a function of its depth. 
If the length of the canal were sufficiently small to per¬ 
mit any disturbance to be propagated back and forth sev¬ 
eral times in the period of the tidal forces, the equilibrium 
theory would apply, and there should be high water at the 
west end at 3 o’clock and at the east end at 9 o’clock, by the 
moon. But, as we have assumed the canal to be half a wave 
length, the equilibrium theory cannot apply, and the phase 
of the oscillation will be, like that of the pendulum, one- 
fourth of a period behind that of the forces; consequently 
the high waters occur at 6 and 12 o’clock, by the moon, at 
the west and east ends, respectively, of the canal. The range 
of the tide depends upon how well the free period of the canal 
agrees with the period of the forces and upon the frictional 
resistance to the motion. 
While there are no such convenient examples as this in 
nature, we may find rough approximations to it. For in¬ 
stance, if we consider a belt or strip of water having its cen¬ 
tral line approximately along the parallel of 10° south latitude 
and extending on either side about 5°, its land boundaries 
will be the Kongo States of Africa on the east and Brazil on 
the west. The middle longitude of this belt is about 13° or 
0.9 hour west. Hence the cotidal hours, considered as de- 
