70 U. S. COAST AND GEODETIC SURVEY 
204. Although by taking a sufficient number of terms the Fourier 
series may thus be made to represent a curve which will be exactly 
satisfied by the n given ordinates, this is, in general, neither necessary 
nor desirable in tidal work, since it is known that the mean ordinates 
obtained from the summations of the hourly heights of the tide in- 
clude many irregularities due to the imperfect elimination of the me- 
teorological effects and also residual effects of constituents having 
periods incommensurable with that of the constituent sought. It is 
desirable to include only the terms of the series which represent the 
true periodic elements of the constituent. With series of observations 
of sufficient length, the coefficient of the other terms, if sought, will 
be found to approximate to zero. 
205. The short-period constituents as derived from the equilibrium 
theory are, in general, either diurnal or semidiurnal. If the period 
of 6 in formula (257) is taken to correspond to the constituent day, 
the diurnal constituents will be represented by the terms with coefficient 
CO, and S,, and the semidiurnal constituents by the terms with co- 
efficients C, and S;. For the long-period constituents, the period of 
6 may be taken to correspond to the constituent month or to the 
constituent year, in which case the coefficients C, and S, wil] refer to 
the monthly or annual constituents and the coefficients C, and S, to 
the semimonthly or semiannual constituents. For most of the 
constituents the coefficients C,, S,, C2, and S, will be the only ones 
required, but for the tides depending upon the fourth power of the 
moon’s parallax and for the overtides and the compound tides, other 
coefficients will be required. Terms beyond those with coefficients 
OC, and S;, for the overtides of the principal lunar constituent are not 
generally used in tidal work. 
206. When it is known that certain periodic elements exist in a 
constituent tide and that the mean ordinates obtained from obser- 
vations include accidental errors that are not periodic, it may be 
readily shown by the method known as the least square adjustment, 
using the observational equations represented by (258), that the most 
probable values of the constant H, and the coefficients C, and S, are 
the same as those given by formulas (285), (289), and (293), 
respectively. 
207. Since in tidal work the value of H,, which is the elevation of 
mean sea level above the datum of observations, is generally deter- 
mined directly from the original tabulation of hourly heights, formula 
(285) is unnecessary except for checking purposes. Formulas (289) 
and (293) are used for obtaining the most probable values of the 
coefficients C, and S, from the hourly means obtained from the 
summations. 
208. When 24 hourly means are used n=24 and u=—15°, and the 
formulas may be written 
] 8=23 
O,=— >5 ha cos li ap (295) 
12 a= 
| 8=23 F 
Ss==5 >) Aasin 15 ap (296) 
12 a= 
in which the angles are expressed in degrees. 
If only 12 means are used, the formulas become 
a=l11 
C=. >> haz cos 30 ap (297) 
a=0 
