HARMONIC ANALYSIS AND PREDICTION OF TIDES 125 
of tides this initial epoch is usually taken at the midnight beginning 
the year for which the predictions are to be made. In strictness the 
V, or uniformily varying portion of the argument alone, refers to the 
initial epoch, while the uw, or slow variation due to changes i in the 
longitude of the moon’s node, is taken as of the middle of the period 
of prediction and assumed to have this value as a constant for the 
entire period. The quantity (V.+) is different for each constituent 
and is also different for each initial epoch and for different longitudes 
on the earth. In table 15 there have been compiled the values o 
this quantity for the beginning of each year from 1850 to 2000 for ie 
the longitude of Greenwich. The values may be readily modified to 
adapt ee to other initial epochs and other longitudes. 
348. Let 
L=west longitude in degrees of station for which predictions 
are desired. 
S=west longitude in degrees of time meridian used at this 
station. 
For east longitude, LZ and S will have negative values. 
Now let 
p=0 when referring to the long-period constituents. 
1 when referring to the diurnal constituents. 
2 when referring to the semidiurnal constituents, etc. 
then p will be the coefficient of the quantity T in the equilibrium 
arguments. Now, 7'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 observation or of prediction. At any given instant of time 
the difference between the values of the hour angle 7 at two stations 
will be equal to the difference in longitude of the stations. If, there- 
fore, the value of the argument (V,+u) for any constituent ‘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 —pL, the negative sign being 
necessary as the value of 7 decreases as the west longitude increases. 
349. The instant of time to which each of the tabular values of 
the Greenwich (V,+1)’s of table 15 refers is the 0 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 0 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 S/15 mean solar hours later than the instant to which 
the tabular Greenwich (V)+1)’s are referred. 
350. In formulas (451) and (452) the symbol a is the general desig- 
nation of the speed of any constituent; that is to say, it is the hourly 
rate of change in the argument. The difference in the argument due 
to a difference of S/15 hours in the initial epoch is therefore aS/15 
degrees. The total correction to the tabular Greenwich (V,+wu) of 
any year in order to obtain the local (V)-+~) for a place in longitude 
L° west at an initial epoch of 0 hours of time meridian S° west at the 
beginning of the same calendar year is 
