38 U. S. COAST AND GEODETIC SURVEY. 



The long-period components are of much less practical importance 

 and only five are usually considered in the analysis. The lunar 

 monthly Mm, with a period of approximately one month indicated 

 by the single s in its argument, ana the lunar fortnightly Mf , and the 

 lunisolar-syncdic fortnightly MSf, with periods of approximately 

 one-half of a month, as indicated by the 2s in their arguments, may 

 be found represented in formula (100). The annual and semiannual 

 components will be referred to later. 



In order to visualize the equilibrium arguncients of the short-period 

 components, the periods of which depend primarily upon the rotation 

 of the earth, it may be found convenient to conceive of a system of 

 fictitious stars, or "astres fictifs," as they are frequently called, 

 which move in the celestial equator. Each diurnal component may 

 be represented by such a star moving at a rate which will cause it to 

 transit the meridian of the place of observation at the instant the 

 argument of the component is zero, the interval between successive 

 transits corresponding to the period of the component. We might 

 conceive the motion of the star relative to the earth's meridian to 

 be strictly uniform corresponding to the rate of change in the V 

 of the argument. In this case the intervals between successive 

 transits will be equal and will determine the length of the mean 

 component day, just as successive transits of the mean sun determined 

 the length of the mean solar day. It may be more convenient, 

 however, to assume that the motion is subject to the inequalities of 

 the u of the argument. In this case the hour angle of the fictitious 

 star will at each instant of time correspond exactly with the argument 

 of the diurnal component that is represented. 



For the semidiurnal components the conception is a little less 

 simple. Perhaps the best assumption is a system of two fictitious 

 stars at 180° apart for each component, moving so that the argument 

 of the component will always be equal to twice the hour angle of 

 either star. Similarly, for the terdiurnal and quarter-diurnal com- 

 ponents, systems of 3 and 4 fictitious stars moving so that the argu- 

 ment of the component is always three or four times, as the case 

 may be, the hour angle of the component star. The conception 

 of the astres fictifs is not adapted to the long-period tides, as these 

 do not depend upon the rotation of the earth for their periods. 



Under the equilibrium theory the time of a component high water 

 will correspond to the zero value of its argument, but under actual 

 conditions the occurrence of a component high water will, in general, 

 be delayed by an amount which is constant for a given place. The 

 lag, expressed in angular measure, is called the epoch of the com- 

 ponent and is usually designated by the Greek letter k. The epochs 

 for any place are determined from actual observations of the tide, 

 and if applied to the equilibrium arguments will give corrected argu- 

 ments which will correspond to the true phases of the component tides 

 at that place. The general expression for the corrected arguments is 

 V+u — K, which will equal zero at the time of the high water of the 

 corresponding component. 



If we adopt some initial instant from which to reckon time, such 

 as the beginning of any series of observations or predictions, and let 



t = number of time units from the initial instant, 



Vo = value of V when t = o, 



a = rate of change in V per unit of time, 



