HARMONIC ANALYSIS AND PREDICTION OF TIDES 7 
The south component is expressed by three terms representing respec- 
tively the long-period, diurnal, and semidiurnal constituents. For the 
west component there are only two terms—the diurnal and semidiur- 
nal, there being no long-period constituents in the west component. 
Each term has been marked separately by a symbol with annexed 
digits analogous to those used for the vertical component to indicate 
the class to which the term belongs. 
83. Comparing formula (90) for the south component with formula 
(32) for the vertical component, it will be noted that the same functions 
of D and ¢ are involved in the corresponding terms of both formulas, 
and that the terms differ only in their numerical coefficient and the 
latitude factor. Allowing for these differences, summarized formulas 
analogous to those given for the vertical component (page 26) may 
be readily formed. In order to eliminate the negative sign of the 
coefficient of the middle term, 180° will be applied to the arguments of 
that term. With all symbols as before, we then have 
F349 /g=9/8 U sn 2Y 2 fC cos EF (92) 
F 3, /g=3/2 U cos 2Y 2 fC cos (E+180°) (93) 
Fx. [g=3/4 U sin 2Y 2 fC cos E (94) 
84. Comparing the two terms in formula (91) for the west com- 
ponent with the corresponding terms in formula (32) for the vertical 
component, it will be noted that the D functions are the same but that 
in (91) the sine replaces the cosine for the functions of ¢. It may be 
shown that the corresponding development of these terms will be 
the same as for the vertical component except that in the developed 
series each argument will be represented by its sine instead of cosine. 
In order that the summarized formulas may be expressed in cosine 
functions, 90° will be subtracted from each argument. With the same. 
symbols as before and allowing for differences in the latitude factors, 
we obtain 
F 3, /g=3/2 Usin Y 2 fC cos (E—90°) (95) 
FPy32 /[g=3/2 U cos Y & fC cos (E—90°) (96) 
85. Formulas for the horizontal component of tide-producing force 
in any given direction may be derived as follows: Let A equal the 
azimuth (measured from south through west) of given direction, and 
let Faso /g, Fos: /g, and Fras. /g, respectively, represent the long-period, 
diurnal, and semidiurnal terms of the component in this direction. 
Then 
F 30 /|g=F sa) |g X cos A (97) 
Fis: /g=F 31 /gXcos A+ Fys: /gXsin A (98) 
ie lg= F 29 /gXcos A+ Fy 39 /gXsin A (99) 
As the long-period term has no west component, the summarized 
formula for the azimuth A may be derived by simply introducing the 
factor cos A into the coefficient of formula (92). For the diurnal and 
semidiurnal terms it is necessary to combine the resolved clements 
from the south and west components. 
86. Referring to formulas (93) to (96) and considering a single 
constituent in each species we. obtain the following: 
