90 U. S. COAST AND GEODETIC SURVEY. 



node) or of the mean sun, and for other short-period components with 

 the transit of a fictitious star representing such component (p. 38). 

 At the point represented by this moon, the angle ( F-l- u) has a value 

 of zero. This angle increases to the right, and at the beginning of the 

 series has a value represented by (Vq + u), which may be readily com- 

 puted for the beginning of any series. This interval from M to the 

 time of occurrence of the first following component high water is the 

 epoch K. This represents the lag or difference between the actual 

 component high water at any place and the theoretical time as 

 determined by the equilibrium theory. The distance from the be- 

 ginning of the series to the following high water is the t of formula 

 (321), which is determined directly from the analysis of the observa- 

 tions. From the figure it is evident that the k is the sum of (Vq + u) 

 and f , and also that it is independent of the time of the beginning of 

 the series. 



Figure 11 gives a more detailed representation of the epoch of a 

 component. In this figure the horizontal line represents changes in 

 time. Distances along this line will be proportional to the changes 

 in the angle of any single component, but since each component 

 has a different speed equal distances along this line will not represent 

 equal angles for different components. The time between the events 

 may be converted into an equivalent component angle by multiplying 

 by the speed of the component. The figure is to some extent self- 

 explanatory. The word "transit" signifies the transit of the fictitious 

 moon representing any component and also the time when the 

 equilibrium argument of that component has a zero value. For all 

 short-period components the time of such zero value will depend 

 upon the longitude of the place of observation as well as upon abso- 

 lute time. For long-period components the zero values are inde- 

 pendent of the longitude of the place of observation, and the ''tran- 

 sits" over the several meridians may be considered as occurring simul- 

 taneously, which is equivalent to taking the coefficient p equal to zero. 

 The figure illustrates the relation between the Greenwich (Vq + u) 

 calculated for the meridian of Greenwich and referring to standard 

 Greenwich time and local (Fq + u) referring to the meridian of 

 observation and the actual time of the beginning of the observations. 



Referring to formulas (100), (208), (215), etc., it will be noted that 

 the element of . each component argument involving the longitude 

 of the place of observation may be represented by ^ T, or p X (hour 

 angle of mean sun), in which p equals the subscript for tne short- 



geriod components and zero for the long-period components. The 

 our angle of the mean sun at any instant is different for each 

 meridian of the earth, the difference being the same as the difference 

 between the longitudes of the places considered. The longitude of 

 Greenwich being zero, the local (V+u) has its origin or zero value 

 at a later time if reckoned toward the west, or at an earlier time if 

 reckoned toward the east, than the Greenwich ( V+ u) . The 

 relation between the local and Greenwich ( F-l- u) may be expressed 



local ( F+ u) = Greenwich {V+u)-p L (339) 



in which Z = longitude of local meridian, positive if west and nega- 

 tive if east. 



The ( Fq + u) is the value of ( V+ u) for a certain specific time. In 

 the reduction of any particular series of observations the local 



