HARMONIC ANALYSIS AND PREDICTION OF TIDES. 163 



The factors F for reduction have been compiled from the formulas 

 indicated in the column of references. The table includes the cor- 

 rected factor for the component M^. The factor in general use for 

 this component is represented by formula (204) on page 52. 



Table 4. Mean longitude of lunar and solar elements. — This table 

 contains the mean longitude of the moon (s), of the lunar perigee (jp), 

 of the sun Qi), of the solar perigee (p^), and of the moon's ascendmg 

 node {N), for January 1, hour, Greenwich mean civil time, for each 

 year from 1800 to 2000, the dates referring to the Gregorian calendar. 



These values are readily derived from Table 2, the rate of change in 

 the mean longitude of the elements for the epoch January 1, 1900, 

 being applicable without material error to any time within the two 

 centuries 1800 to 2000 covered by Table 4. The same rate of change 

 may also be used, without introducing any errors of practical impor- 

 tance, to extend Table 4 to dates beyond these limits. In extending 

 the table, care should be taken to distinguish between the common 

 and leap years, and for the earlier dates due consideration should be 

 given to the kind of calendar in use. (See p. 11 for discussion of 

 calendars.) It will be noted that each Julian century contains 36,525 

 days, while the common Gregorian century contains only 36,524 days^ 

 with an additional day every fourth century. 



Table 5. Differences to adapt Table 4 to any month, day, and hour. — 

 These differences are derived from the daily and hourly rate of change 

 of the elements as given in Table 2, multiples of 360° being rejected 

 when they occur. The table is prepared especially for common years, 

 but is appHcable to leap years by increasing the given date by one 

 day if it is between March 1 and December 31, inclusive. The cor- 

 rection for the hour of the day refers to the Greenwich hour, and if 

 the hour for which the elements are desired is expressed in another 

 kind of time the equivalent Greenwich hour must be used for the 

 table. 



Table 6. Values of I, v, ^, v' , and 2v" for each degree of N. — This 

 table has been computed for epoch January 1, 1900, using the con- 

 stants of Table 2, but the tabular values are applicable without 

 material error to any series of observations within modern times. 



The following formulas were used in the computations for this 

 table: 

 Cos /=cos i cos CO — sin -i sin CO cos N 



= 0.91369-0.03569 cos N (103) of p. 41. 



Tan ^(iV-? + v)=^^^4fe4 tan ^N (104) of p. 41 

 ^ cos i(co + ^) ^ ^ ^ r 



Tan HN-^-v) =|^||^ tan Jif (105) of p. 41 



Log ^S4r^ -0.00810; log ^!° ti" I 'i - 9.80897 

 ^ cos •J(co^-^) ' ^ sm ^(co + i) 



m , sin V sin 2/ 



Tan v' = 



COS V sin 27+ ^ , ^l ( - j ^ sin 2co 



2 + 3e^ 



2 + 3e2 \cj M' 



sin V sin 2/ ,^„_. „ 



COS. sin 27+ 0.3357 (229) of p. 57 



