HAKMONIC ANALYSIS AND PREDICTION OF TIDES. 101 



any multiple thereof, or, in other words, as r approaches in length 

 any multiple of the synodic period of components A and B. Also, 

 since the numerator of the fraction can never exceed unity, while the 

 denominator may be increased indefinitely, tne value of the fraction 

 will, in general, be diminished by increasing the length of series and 

 will approach zero as r approaches infinity. The greater the dif- 

 ference (b — a) between the speeds of the two components the less 

 will be their disturbing effects upon each other. For this reason the 

 effects upon each other of the diurnal and semidiurnal components 

 or of any components of difi'erent subscripts is usually considered as 

 negligible, and in the application of formulas (409) and (410) only 

 components with like subscripts are taken into account. 



The quantities R{B) and f(5) of formulas (409) and (410) refer 

 to the true amplitudes and epochs of the disturbing components. 

 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 cover 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 components 

 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 components should be used. 



31. LONG-PERIOD COMPONENTS. 



The preceding discussions have been especially applicable to the 

 reduction of the short-period components — those having a period of 

 a component day or less. They are the components that determine 

 the daily or semidaily rise and fall of the tide. Consideration will 

 now be given to the long-period tides which afi'ect 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 components that are usually treated in works on 

 harmonic analysis — the lunar fortnightly Mf, the lunisolar synodic 

 fortnightly MSf, {he 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 

 meteorological conditions, often have an appreciable efi'ect upon the 

 mean daily level of the water. 



To obtain the long-period components, methods similar to those 

 adopted for the short-period components with certain modifications 

 may be used. For the fortnightly and monthly components the 

 component month may be divided into 24 equal parts, analogous 

 to the 24 component hours of the day. Similarly, for the semiannual 

 and annual components the component year may be divided into 

 24 equal parts, although it will often be found more convenient to 

 divide the year into 12 parts to correspond approximately with the 

 12 calendar months. 



Instead of distributing the individual hourly heights, as for the 

 short-period components, a considerable amount of labor can be 

 saved by using the daily sums of these heights. The mean of each 



