4 U. S. COAST AND GEODETIC SURVEY. 



Figure 1 is a graphic representation of a component tide and illus- 

 trates certain relations oi quantities given in the above formulas. 

 In this figure the full horizontal line represents the mean level of the 

 component, and distances along this line correspond to time as meas- 

 ured by an angle which increases uniformly with time. The height 

 of the tide at any time is represented by an ordinate to the curve 

 perpendicular to the mean level line. The height of the maximum 

 or high water of the component is the coefficient fH of the formulas. 



The point 31 in the figure indicates the instant of time at which 

 high water would occur in accordance with the uncorrected equilib- 

 rium theory. The epoch k is the angular expression for the interval 

 between the time of the theoretical high water and the actual high 

 water of the component as determined from observations. The in- 

 terval between two consecutive high waters is the period of the com- 

 ponent and is represented by the angular cycle of 360°. The interval 

 measured backward from M to the preceding high water may there- 

 fore be expressed by 360° — /c or — k. 



Let the vertical line through the point T indicate any instant of 

 time t under consideration. Then the interval between M and this 

 instant will correspond to the angle (V+u) of formula (4). If this 

 instant represents the initial epoch from which the time is to be 

 reckoned, the (V+u) becomes (Vo + u) of formula (5), and the in- 

 terval from T to the time of the following high water becomes the 

 ^ of formula (3). The interval measured backward from T to the 

 preceding high water is the a of formula (2) . 



The epoch k equals the sum of the {Vo + u) and C- 



3. EQUILIBRIUM THEORY. 



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. A 

 surface of equilibrium, also known as an equipotential or level sur- 

 face, is a surface at every point of which the potential has the same 

 value; that is to say, the potential energy of a particle in such a sur- 



