28 U. 8S. COAST AND GEODETIC SURVEY 
Diurnal constituent, 
3/2 UfC [cos 2Y cos A cos (H+180°)+sin Y sin A cos (H—90°)] 
=3/2 UfC (—cos 2Y cos A cos H-+sin Y sin A sin E) 
=3/2 UfC P, cos (E—X,) (100) 
in which 
P,=(cos? 2Y cos? A+sin? Y sin? A)} (101) 
ea sin Y sin A (102) 
—cos 2Y cos A 
Semidiurnal constituent, 
3/2 UfC [sin Y cos Y cos A cos E-+cos Y sin A cos (H—90°)] 
=3/2 UfC cos Y (sin Y cos A cos E-+sin A sin EF) 
=3/2 UfC P, cos (H—X2) (103) 
in which 
P,=cos Y (sin? Y cos? A+sin? A)? (104) 
eae (105) 
sin Y cos A 
87. Summarized formulas for the horizontal component of the 
tide-producing force in any direction A may now be written as follows: 
F 3) /g=9/8 U sin 2Y cos A 2 fC cos EF (106) 
Fon, /g—3/2 UP, = fC cos (H—X,) (107) 
Fz. /[g=3/2 UP, = fC cos (H— X2) (108) 
the values for P,, P:, X; and X, being obtained by formulas in the 
preceding paragraph. P, and P, are to be taken as positive and the 
following table will be found convenient in determining the proper 
quadrant for X, and X). 
North latitude South latitude 
IN —_——— ———<——— —- 
quadrant Xi Xp Xi Xs 
quadrant quadrant quadrant quadrant 
1 2 
2 1 or 2 
3 4or3 
4 3 
For the X; quadrant the first value of each pair is applicable when the latitude does not exceed 45° north 
orsouth. Otherwise the second value is applicable. 
EQUILIBRIUM TIDE 
88. The equilibrium theory of the tides is a hypothesis under which 
it is assumed that the waters covering the face of the earth instantly 
respond to the tide-producing forces of the moon and the sun and 
form a surface of equilibrium under the action of these forces. The 
theory disregards fricticn and inertia and the irregular distribution of 
the land masses of the earth. Although the actual tidal movement 
