HARMONIC ANALYSIS AND PREDICTION OF TIDES. 121 



The values may be readily modified to adapt them to other initial 

 epochs and other longitudes. 

 Let 



jC==west longitude in degrees of station for which predictions 



are desired. 

 5' = west longitude in degrees of time meridian used at this 



station. 



For east longitude, L and S will have negative values. 



Now let 



2? = when referring to the long-period components. 



1 when referring to the diurnal components. 



2 when referring to the semidiurnal components, etc. 



then 2> will be the coefficient of the quantity T in the equilibrium 

 arguments represented in (100), (215), and other formulas. Now, T 

 is the hour angle of the mean sun and is the only quantity in these 

 arguments that is a function of the longitude of the place of observa- 

 tion or of prediction. At any given instant of time the difference 

 between the values of the hour angle T at two stations will be equal 

 to the difference in longitude of the stations. If, therefore, the value 

 of the argument (Fo + u) for any component at any given instant 

 has been computed for the meridian of Greenwich, the correction to 

 refer this argument for the same instant to a place in longitude L° 

 west of Greenwich will be — fL, the negative sign being necessary as 

 the value of T decreases as the west longitude increases. 



The instant of time to which each of the tabular values of the 

 Greenwich {Vo + uYs, of Table 15 refers is the hour of the Green- 

 wich mean civil time at the beginning of a calendar year. In the 

 predictions of the tides at any station it is desirable to take as the 

 initial epoch the hour of the standard or local time customarily 

 used at that station. If, therefore, the longitude of the time merid- 

 ian used is S° west of Greenwich, the initial epoch of the predictions 

 will usually be Sjlb mean solar hours later than the instant to which 

 the tabular Greenwich {Vq + uYs are referred. 



In formulas (465) and (466) the symbol a is the general designation 

 of the speed of any component; that is to say, it is the hourly rate of 

 change in the argument. The difference in the argument due to a 

 difference of Sjlb hours in the initial epoch is therefore aSjlb degrees. 

 The total correction to the tabular Greenwich {Vo + u) of any year in 

 order to obtain the local {Vo + u) for a place in longitude U' west at 

 an initial epoch of hours of time meridian S° west at the beginning 

 of the same calendar year is 



ff-pi- (467) 



The general expression for the angles of (465) and (466) may now 

 be written 



[at +{Vo + u)-K'\ = [at + Greenwich {Vo + u) +^-'pL-K\ (468) 



In order to avoid the necessity of applying the corrections for longi- 

 tude and initial epoch to the Greenwich {V^ + uY^ for each year, 

 these corrections may be applied once for all to the k's 



