HARMONIC ANALYSIS AND PEEDICTION OF TIDES. 



39 



then we may write 



iV+u-K) = {at+ Vo + u-k) 



(101) 



In (101) u will be assigned a value corresponding to the middle 

 of the series under consideration and will be assumed to hold that 

 value as a constant for the entire series. 



Table 3 contains the formulas for the arguments of the principal 

 components and also the hourly rates of change in the argument, 

 and Table 15 gives the values of Fo + w for the meridian of Greenwich 

 for the beginning of each year from 1850 to 2000 as computed from 

 the formulas. 



A graphical representation of the relations between Vo + u, f, and 

 K is shown in Figure 11. The heavy horizontal line represents the 

 time argument advancing to the right, the distance being expressed 

 in angular measurement that increases uniformly with the advance 

 of time. The figure takes account of a single typical short-period 

 component with an hourly speed of a, the ratio of this speed to that 



J 



1^ 



SI 



■ (Treenw/icb V,+u 



• pL. 



.1 n 





^s 



,s_T>(S-L). 







-cS- 

 -cL- 



c(S-g-^ 



— — KRefcrredtoTi'me rteridia.T) Used 



-Local V,+u X 4- 



-T^ue K- 



Timri > 



Fig. U. 



of the mean sun being represented by c. In the figure the horizontal 

 distance that corresponds to one hour in time is equivalent to a 

 units of the angular measurement. 



The point indicated as "0^ of time used for observations" is 

 assumed to represent the exact beginning of the series of observations 

 analyzed; that is, the time of the first hourly height of the tabulations. 

 The interval between the beginning of the observations and the 

 time of the first following component high water is indicated by f. 

 The interval between the next preceding transit of the astre fictif 

 over the local meridian and the beginning of the series is designated 

 as the local Vo + u. The true epoch or k is the interval between the 

 transit of the astre fictif over the local meridian and the time of the 

 following component high water, and therefore equals the sum of 

 i he local Vo + u and the f . 



The Greenwich Vo + u as given in Table 15 is the interval between 

 the transit of the astre fictif over the meridian of Greenwich and the 

 hour of the following Greenwich day. The interval between the 

 transit over the meridian of Greenwich and the transit over any other 

 meridian is equal to the product of the subscript of the component 

 and the difference in longitude, the subscript indicating the number 

 of component periods in a component day. For east longitude the 

 transit would occur earlier and for west longitude later than the 



