40 U. S. COAST AND GEODETIC SURVEY. 



transit over the meridian of Greenwich. For the long-period com- 

 ponents an initial epoch of the Fo + w is reckoned from certain astro- 

 nomical relations independent of the rotation of the earth and is 

 consequently independent of longitude on the earth. Therefore, to 

 adapt Figure 11 to the long-period component, the subscript p must 

 be assumed to be zero. 



The change in the Vo + u from the hour of a Greenwich day to the 

 hour of the same calendar day defined by another time meridian 

 equals the product of the speed ratio c and the difference in longitude 

 of the time meridians. 



From the figure it is evident that a correction of ( — pL + cS) must 

 be applied to the Greenwich Vo + u to obtain the local Vo-\-u. 



If the epochs or ks are referred to some standard time meridian 

 instead of the local meridian at the place of observations, a correction 

 equal to the product of the component subscript and the difference 

 between the longitude of the local and standard meridians must be 

 applied to reduce such epochs to the local meridian, the subscript 

 being taken as zero for all the long-period components. 



11. COEFFICIENTS. 



Kef erring to formula (100), on page 35, it wiU be noted that 

 the coefficients of the terms are made up of two parts — a general 

 coefficient applying to all the terms within a group and an individual 

 coefficient applying to a single term. In this formula three groups 

 of components are represented — the semidiurnal, the diurnal, and the 

 long-period tides. The general coefficient of each group includes the 



common factor 3/2 -^ ( — ) a, but differs from the others in the factor 



involving the latitude (X) of the place of observation. The individual 

 coefficients of the components of a single group are therefore directly 

 comparable with each other and will give the relative theoretical 

 importance of the components of that group. The relative import- 

 ance of the components in different groups wiH depend also upon the 

 latitude of the place of observation. For the semidiurnal compo- 

 nents the general coefficient will have a maximum value for places 

 on the Equator; for the diurnal tides in latitude 45° north or south; 

 and for the long-period tides at the north and south poles. For 

 convenience the term coefficient is frequently applied to the indi- 

 vidual coefficient, exclusive of the general factor, and may be so 

 used in the following discussion of mean values. 



A general expression for each term in equation (100) is 



J cos (F-fw) (102) 



in which the coefficient J is a function of /, and u is a function of p 

 and f. Since /, v, and 6 are all functions of the longitude of the 

 moon's node, which is usually represented by N, the values of / 

 and u will also be functions of N. If we assume a succession of a 

 great many short series of tidal observations to be analyzed, the mean 

 of the resulting amplitudes for any component might be represented 

 by the mean value of J in (102) ; but if a single very long continuous 

 series is to be analyzed the resulting amplitude will be more accurately 

 represented by the mean value of the product J cos u. The difference 

 may be explained as follows: The inequalities due to the u in the 



