HARMONIC ANALYSIS AND PREDICTION OF TIDES 25 
For terms A,, to A,, in formula (63) 
[sin J cos? 4] cos (2—v)]p>=sin w cos’ tw cos* 44=0.3800 (67) 
For terms Ay»: to Ajo in formula (63) 
[sin 27 cos y]>=sin 2a (1—3/2 sin? 2) =0.7214 (68) 
For terms A3, to A; in formula (63) | 
[sin J sin? 4] cos (2+ y)]p>=sin w sin’ 4w cos* 42=0.0164 (69) 
For terms Aj) to Ax, in formula (64) 
[cos 4] cos (2&—2v)]o>=cos* 4m cos* 41=0.9154 (70) 
For terms Ay, to A;; in formula (64) 
[sin? J cos 2v]>=sin? w (1—3/2 sin? 7) =0.1565 (71) 
For terms A;_ to Ags in formula (64) 
[sin* 47 cos (2+2y)],>=sin* $w cos* 31 =0.0017 (72) 
77. The ratio obtained by dividing the true obliquity factor for 
any value of J by its mean value may be called a node factor since it is 
a function of the longitude of the moon’s node. The symbol generally 
used for the node factor is the small f. The node factor may be used 
with a mean constituent coefficient to obtain the true coefficient 
corresponding to a given longitude of the moon’s node. Node factors 
for the several terms of formulas (62) to (64) may be expressed by the 
following ratios: 
f(A)) to f(A;) =f(Mm) = (2/3—sin? 1)/0.5021 (73) 
f(As) to f(A:3) =f(Mf) =sin? I /0.1578 (74) 
F(Ays) to f(A) =f(O1) =sin I cos? J /0.3800 (75) 
f(Asy) to f( Ag) =f(J:) =sin 2T /0.7214 (76) 
f(Az;) to f(Ass) =f(OO,)=sin J sin? 4] /0.0164 (77) 
(Aso) to f(Asg) =f(M2) =cos! 41 /0.9154 (78) 
f(Ayz) to f(A55) sin? I /0.1565 (79) 
f(Age) to f(Ags)=sin! 4 I /0.0017 (80) 
Node factors for the middle of each calendar year from 1850 to 1999 
are given in table 14 for the constituents used in the Coast and 
Geodetic Survey tide-predicting machine. These include all the 
factors above excepting formulas (79) and (80). However, since 
formula (79) represents an inerease of only about one per cent over 
formula (74), the tabular values for the latter are readily adapted to 
formula (79). Node factors change slowly and interpolations can be 
made in table 14 for any desired part of the year. For practical 
purposes, however, the values for the middle of the year are generally 
taken as constant for the entire year. 
78. The reciprocal of the node factor is called the reduction factor 
and is usually represented by the capital F. Applied to tidal coeffi- 
cients pertaining to any particular year, the reduction factors serve 
to reduce them to a uniform standard in order that they may be 
comparable. Logarithms of the reduction factors for every tenth of 
a degree of J are given in table 12 for the constituents used on the 
tide-predicting machine of this office. 
79. Formulas (62), (63), and (64), for the long-period, diurnal, and 
semidiurnal constituents of the vertical component of the tide-pro- 
ducing force may now be summarized as follows: 
Let H=constituent argument from table 2 
C=mean constituent coefficient from table 2 
f =node factor from table 14 
