HARMONIC ANALYSIS AND PREDICTION OF TIDES 87 
4(b—a)r will depend upon the length of series; but for any given 
length of series they will be constant for all times and places. Table 
29 has been computed to give these quantities for different lengths 
of series. The factor as directly obtained may be either positive or 
negative, but for convenience the tabular values are all given as 
positive, and when the factor as directly obtained is negative the 
angle has been modified by +180° in order to compensate for the 
change of sign in the factor and permit the tabular values to be used 
directly in formulas (389) and (390). _ 
252. An examination of formulas (889) and (390) will show that the 
disturbing effect of one constituent upon another will depend largely 
sin $(b—a)r 
4(b—a)r 
not equal to a, this fraction and the disturbing effect it represents will 
= ° 
approach zero as the length of series 7 approaches in value a or 
upon the magnitude of the fraction Assuming that b is 
any multiple thereof, or, in other words, as 7 approaches in length 
any multiple of the synodic period of constituents A and B. Also, 
since the numerator of the fraction can never exceed unity, while the 
denominator may be increased indefinitely, the value of the fraction 
will, in general, be diminished by increasing the length of series and 
will approach zero as 7 approaches infinity. The greater the dif- 
ference (b—a) between the speeds of the two constituents the less 
will be their disburbing effects upon each other. For this reason the 
effects upon each other of the diurnal and semidiurnal constituents 
are usually considered as negligible. 
253. The quantities R(B) and ¢(B) of formulas (389) and (390) refer 
to the true amplitudes and epochs of the disturbing constituents. 
These true values being in general unknown when the elimination 
process is to be applied, it is desirable that there should be used in the 
formulas the closest approximation to such values as are obtainable. 
If the series of observations covers a period of a year or more, the am- 
plitudes and epochs as directly obtained from the analysis may be 
considered sufficiently close approximations for use in the formulas. 
For short series of observations, however, the values as directly 
obtained for the amplitudes and epochs of some of the constituents 
may be so far from the true values that they are entirely unservice- 
able for use in the formulas. In such cases inferred values for the 
disturbing constituents should be used. 
LONG-PERIOD CONSTITUENTS 
254. The preceding discussions have been especially applicable to 
the reduction of the short-period constituents—those having a period 
of a constituent day or less. They are the constituents that deter- 
mine the daily or semidaily rise and fall of the tide. Consideration 
will now be given to the long-period tides which affect the mean level 
of the water from day to day, but which have practically little or no 
effect upon the times of the high and low waters. There are five 
such long-period constituents that are usually treated in works on 
harmonic analysis—the lunar fortnightly Mf, the lunisolar synodic 
fortnightly MSf, the lunar monthly Mm, the solar semiannual Ssa, 
and the solar annual Sa. The first three are usually too small to be 
of practical importance, but the last two, depending largely upon 
