HAEMONIC AE^ALYSIS AND PREDICTIOlsr OF TIDES. 



83 



By substituting (306), (310), (311), and (314) in (278), the follow- 

 ing equation of a curve, which will pass through the n given points,. 

 wiU be obtained 



I a=(n-l) 

 n a=o 



"2 a=(n-l) 



- S /^a cos a u 



Ji' a=o 



"2 a=(ii-l) _ 



- S '^a. sm a u 

 _n a=o 

 "2 a=(n-l) 



- S ^a cos 2 

 n a=o 

 "2 a=(n-l) -| 



- S ^a sin 2 a w sin 2 



l_7l. a=o J 



auj 



cos^ 

 sin d 

 cos 2 ^ 



"2* a=(n-l) -j 



— S '^a COS ^ a tt COS Ic 

 n a=o J 



"2 a=(ii-l) -I 



— S ^ia sin I a u \ sin Z 



_7l a=o _| 



(315) 



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 components having 

 periods incommensurable with that of the component sought. It is 

 desirable to include only the terms of the series which represent the 

 true periodic elements of the component. With series of observations 

 of sufficient length, the coefficient of the other terms, if sought, will 

 be found to approximate to zero. 



By a reference to formula (100), page 35, it will be noted that the* 

 short-period components as derived from the equilibrium theory are,, 

 in general, either diurnal or semidiurnal. If the period of 6 in formula 

 (278), page 76, is taken to correspond to the component day, the- 

 diurnal components will be represented by the terms with coefficient 

 Ci and Si, and the semidiurnal components by the terms with 

 coefficients C^ and 82- For the long-period components, the period of 

 6 may be taken to correspond to the component month or to the- 

 component year, in which case the coefficients C^ and S^ will refer to 

 the monthly or annual components and the coefficients Cj and S2 to- 

 the semimonthly or semiannual components. 



For most of the components the coefficients C^, S^, C^, and S^ will 

 be the only ones required, but for the tides depending upon the fourth, 

 power of the moon's parallax (sec. 15) for the overtides (sec. 18), and 

 the compound tides (sec. 19) , other coefficients will be required. Terms, 

 beyond those with coefficients Cg and Sg, for the overtides of the prin- 

 cipal lunar component, are not generally used in tidal work. 



When it is known that certain periodic elements exist in a component 

 tide and that the mean ordinates obtained from observations include 



* If 71 is even and fc=- 



