62 U. S. COAST AND GEODETIC SURVEY 
diurnal, 45° for sixth-diurnal, and 60° for eighth-diurnal constituents. 
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 
constituent 2Q when based upon the primary summations for O. 
This is asmall and unimportant constituent, and heretofore no analysis 
has been made for it, the value of its harmonic constants being in- 
ferred from those of constituent 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 constituent which will be more satisfactory than the inferred 
constants. 
FOURIER SERIES 
187. 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 
h=H,+C, cos 6+ ©, cos 26+ 0; cos 36+ ______ 
+S, sin 6+ S, sin 20+S; sin 36+______ 
It can be shown that by taking a sufficient number of terms the 
Fourier series may be made to represent any periodic function of 6. 
This series may be written also in the following form: 
h=H)+ A; cos (8+ a;)+ Az cos (20+ a,)+ Az cos (80+ a3) +__-._ (255) 
in which 
(254) 
An=[Cn?+Sn’|? and an=-—tan! = 
m being the subscript of any term. 
188. From the summations for any constituent 24 hourly 
means are obtained, these means being the approximate heights 
of the constituent tide at given intervals of time. These mean 
constituent hourly heights, together with the intermediate heights, 
may be represented by the Fourier series, in which 
Hy=mean value of the function corresponding to the height of 
mean sea level above the adopted datum. 
é=an angle that changes uniformly with time and completes a 
cycle of 360° in one constituent day. The values of 6 corresponding 
to the 24 hourly means will be 0°, 15°, 30°, _ _ _ - 330°, and 345°. 
Formula (254), or its equivalent (255), is the equation of a curve 
with the values of @ as the abscisse and the corresponding values of 
h as the ordinates. If the 24 constituent hourly means are plotted as 
ordinates corresponding to the values of 0°, 15°, 30°, _ _ _ — for @, 
it is possible to find values for Hy, Cn, and Sm, which when substituted 
in (255) will give the equation of a curve that will pass exactly through 
each of the 24 points representing these means. 
189. In order to make the following discussion more general, let it be 
assumed that the period of 6 has been divided into n equal parts, and 
that the ordinate or value of A pertaining to the beginning of each of 
those parts is known. Let w equal the interval between these ordi- 
nates, then 
NM U=27, or 360° (256) 
Let the given ordinates be ho, hi, hy ---- h «1 corresponding 
to the abscissae 0, u, 2u ____ (n—1) u, respectively. 
