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76 U. S. COAST AND GEODETIC SURVEY 
its phase or value pertaining to an initial instant of time, such as the 
beginning of a series of observations, is expressed by (V,+u). Refer- 
ring to formula (300), since @ is reckoned from the beginning of the 
series, the angular quantity (—{) is the corresponding phase of the 
observed constituent at this time. The phase lag may therefore be 
expressed by the following general formula: 
k= Votu—(—-p)=Votute (316) 
222. Since the argument formulas of all short-period constituents 
contain some multiple of the hour angle (7) of the mean sun, the 
arguments themselves will have different values in different longitudes 
at the same instant of time. If p equals the coefficient_of 7 or the 
subscript_of the constituent-and LZ equals the longitude of the place 
“in degrees reckoned west from Greenwich, L being considered as nega- 
tive for east longitude, the relation between the local and Greenwich 
argument for any constituent may be expressed as follows: 
local (V-++-uv)=Greenwich (V+u)—pL (317) 
223. Also, since the absolute time of the beginning of a day or 
the beginning of a year depends upon the time meridian used in the 
locality, the mitial instant taken for the beginning of a series of obser- 
vations may differ in different localities even though expressed in the 
same clock time of the same calendar day. If we let S equal the 
longitude of the time meridian in degrees, positive for west and nega- 
tive for east, the same meridian expressed in hours becomes S/15. 
Letting a equal the speed or hourly rate of change in the constituent 
argument, the difference in argument due to the difference in the 
absolute beginning of the series becomes a@S/15, and the relation 
between the local and Greenwich argument due to this difference 
may be expressed as follows: 
local (V,+u)=Greenwich (V,+u)—pL-+aS/15 (318) 
In the above formula the local and Greenwich (V,+«) pertain to the 
same clock time but not the same absolute time unless both clocks are 
set for the meridian of Greenwich. 
224. Values of (V.+wu) for the meridian of Greenwich at the 
beginning of each calendar year 1850 to 2000 are given in table 15 
for all constituents represented in the Coast and Geodetic Survey 
tide-predicting machine. Tables 16 to 18 provide differences for 
referring the arguments to other days and hours of the year. In the 
preparation of table 15 that portion of the argument included in the u 
was treated as a constant with a value pertaining to the middle of the 
calendar year. If the Greenwich (V,-++v) with its corrections is sub- 
stituted for the local (_V,+ wu) in formula (316), we obtain 
x=Greenwich (V,+u)—pL+aS/15+¢ (319) 
225. The phase lag designated by « is sometimes called the local 
epoch to distinguish it from certain modified forms which may be used 
for special purposes. In the preparation of the harmonic constants 
for predictions it is convenient to combine the longitude and time 
meridian corrections with the local epoch to form a modified epoch 
