28 



tion. From the discussion in paragraph 46 it is clear that the speed, 

 a, of each component is determined by the astronomical movements 

 of the moon or sun, or both. The amplitude A, and the initial phase 

 a may be determined from the recorded tides at the place, by a 

 method hereinafter described. In the ensuing discussion, the ampli- 

 tude A will be expressed in feet ; the time t and the period T, in mean 

 solar hours; the speed, a, in degrees per hour (unless otherwise indi- 

 cated) and the initial phase a in degrees. 



Each component is graphically represented by the projection on the 

 Y axis (fig. 16), of a generating radius CP, of length equal to the 



Figure 13. —Generating radius and tide curve of a component. 



amplitude of the component, rotating at the constant speed of the 

 component around the origin, C, its initial angle with the Y axis being 

 the initial phase of the component. 



The direction of the rotation of the generating radius is taken as 

 positive in a counterclockwise direction, in accordance with the usual 

 trigonometric convention. 



The graph of the component is the sinusoidal cosine curve shown 

 on the right in figure 16, in which the abscissas represent time and the 

 ordinates the height of the component above mean sea level. 



It is sometimes more convenient to write equation (27) in the form: 



y=HQ+A^ cos (ai^-ri)+^2 cos (aoi-r2)+^3 cos (a3^-f3)+ ' • • (29) 



in which the angles designated as f (zeta) are numerically equal to the 

 respective initial phases, a, of the several components but opposite 

 in sign. Since each component reaches its maximum when at — f =0, 

 and hence when t = ^/a, it is evident that f/a is the time of high water 

 of the component next after the origin of time. 



50. "Astres Fidijs." — The tide represented by a single component 

 would have high waters and low waters of constant heights occurring 

 at equal intervals of time. Such a tide would be generated by a 

 moon traveling at a constant angular speed along a circular orbit in 

 the plane of the earth's Equator. Each component of the tide is 

 therefore sometimes treated as the tide due to a fictitious moon, or 

 "astre fictif," moving at a uniform speed along the earth's celestial 



