HAHMONIC AINALiYSIS AND PKEDICTIOlSr OF TIDES. 75 



In the ordinary primary summation the extreme difference between 

 the time of the observation of a solar hourly height and the intregal 

 component hour to wnich it is assigned is one-half of a component 

 hour and, represented by component degrees, it is 7.5° for diurnal, 15° 

 for semidiurnal, 22.5° for terdiurnal, 30° for quarter diurnal, 45° for 

 sixth-diurnal, and 60° for eighth-diurnal components. By the above 

 schedule it will be noted that the extreme difference exceeds 60° 

 in only a few cases. The largest difference is 99° for component 2Q 

 when based upon the primary summations for O. This is a small and 

 unimportant component, and heretofore no analysis has been made 

 for it, the value of its harmonic constants being inferred from those of 

 component O. Although theoretically too small to justify a primary 

 summation in general practice, the lesser work involved in the 

 secondary summations may produce constants for this component 

 which will be more satisfactory than the inferred constants. 



Although the general use of secondary stencils for series of obser- 

 vations less than six months in length is not at present recommended, 

 it is possible that future tests may indicate that these stencils may 

 he used to advantage with shorter series. 



26. THE FOURIER SERIES. 



A series involving only sines and cosines of whole multiples of a 

 varying angle is generally known as the Fourier series. Such a series 

 is of the form 



Ti, = ^o + C'l cos + (72 cos 20 + Cg cos 30 + .... 



+ ;SiSin0 + /S'2 sin 20 + iS3 sin 30+ ■ • • • ^ ' 



It can be shown that by taking a sufficient number of terms the 

 T'ourier series may be made to represent any periodic function of d. 

 This series may be written also in the following form : 



Ti = E^ + A^ cos (0 + o!i) + J.2 cos (20 + a^) + A^ cos (30 + 0:3) H (276,) 



in which 



A^ = [C^' + S^'f and «„.= -tan-^f^ 



m being the subscript of any term. 



From the summations for any component 24 component hourly 

 means are obtained, these means being the approximate heights of 

 the component tide at given intervals of time. These mean com- 

 ponent hourly heights, together with the intermediate heights, may 

 be represented by the Fourier series, in which 



-ffo = mean value of the function corresponding to the height of 

 jnean sea level above the adopted datum. 



= an angle that changes uniformly with time and completes a 

 cycle of 360° in one component day. The values of corresponding 

 to the 24 hourly means will be 0°, 15°, 30°, • • • • 330°, and 345°. 



Formula (275), or its equivalent (276), is the equation of a curve 

 ivith the values of as the abscissse and the corresponding values of 

 A as the ordinates. If the 24 component hourly means are plotted as 

 ordinates corresponding to the values of 0°, 15°, 30°, .... for 0, it is 



72934— 24t 6 



