HARMONIC ANALYSIS AND PREDICTION OF TIDES 33 
= (M/E) (a/d)*(5 cos? z—3 cos 2) (121) 
Letting 7=a-+A, and expanding the first member of the above formula, 
it becomes equal to A, /a after the rejection of the higher powers of this 
small fraction. The formula may then be written 
hy /a=1/2 (M/E) (a/d)*(5 cos? z—3 cos 2) (122) 
99. Formulas (119) and (122) involving the cube and 4th power of 
the moon’s parallax, respectively, represent the equilibrium heights 
of the tide due to the corresponding forces, the heights being expressed 
in respect to the mean radius (a) of the earth as the unit. In deriving 
these formulas the centrifugal force of the earth’s rotation was dis- 
regarded and the resulting heights represent the disturbances in a 
true spherical surface due to the action of the tide-producing force. 
It may be inferred that in a condition of equilibrium the tidal forces 
would produce like disturbances in the spheroidal surface of the earth 
and the A of the formulas may therefore be taken as being referred to 
the earth’s surface as defined by the mean level of the sea. 
100. The extreme limits of the equilibrium tide, applicable to the 
time when the tide-producing body is nearest the earth, may be 
obtained by substituting the proper numerical values in formulas 
(119) and (122). They are given below for both moon and sun. 
From formula (119) involving the cube of parallax— 
Greatest rise =1.46 feet for moon, or 0.57 foot for sun (123) 
Lowest fall =0.73 foot for moon, or 0.28 foot for sun (124) 
Extreme range=2.19 feet for moon, or 0.85 foot for sun. (125) 
From formula (122) involving the 4th power of parallax— 
Greatest rise =0.026 foot for moon, or 0.000025 foot for sun (126) 
Lowest fall | =0.026 foot for moon, or 0.000025 foot for sun Gi2 7) 
Extreme range=0.052 foot for moon, or 0.00005 foot for sun. (128) 
101. A comparison of formulas (23) and (119), the first expressing 
the relation of the vertical component of the principal tide-producing 
force to the acceleration of gravity (g) and the other the relation of 
the height of the corresponding equilibrium tide to the mean radius (a) 
of the earth, will show that they are identical with the single excep- 
tion that the coefficient of the height formula is one-half that of the 
force formula. Therefore the development of the force formula into 
a series of harmonic constituents is immediately applicable in obtain- 
ing similar expressions for the equilibrium height of the tide. Using a 
notation for the height terms corresponding to that used for the force 
terms, let hg /a, hz; /a, and hg, /a represent, respectively, the long- 
period, diurnal, and semidiurnal terms of the equilibrium tide involv- 
ing the cube of the moon’s parallax. Then referring to formulas (81) 
to (83) we may write 
hz /a=3/4 UA/2—3/2 sin? Y) 2 fC cos E (129) 
hs, /4=3/4 U sin 2Y 2 fC cos E (130) 
hz [a=3/4 U cos? Y 2 fC cos E (131) 
the symbols having the same significance as in the preceding discussion 
of the tidal forces. 
