HARMONIC ANALYSIS AND PREDICTION OF TIDES 7 
the subscript o being used for the long-period constituents. In 
formula (32) the individual terms are identified by the annexation 
of the species subscript to the general symbol for the formula. 
53. As written, all of the three terms of formula (32) have the 
same coefficient 3/2 (M/E) (a/d)?. In each case the latitude (Y) 
factor has a maximum value of unity, this maximum being negative 
for the first term. For the long-period term (F'2) /g), the latitude 
factor has a maximum positive value of }4 at the equator, becomes zero 
in latitude 35.26° (approximately), and reaches a maximum negative 
value of—1 at the poles, the factor being the same for corresponding 
latitudes in both northern and southern hemispheres. For the diurnal 
term (F’,3; /g), the latitude factor is positive for the northern hemisphere 
and negative for the southern hemisphere. It has a maximum 
value of unity in latitude 45° and is zero at the equator and poles. 
For the semidiurnal terms (F’,3. /g), the latitude factor is always posi- 
tive and has a maximum value of unity at the equator and equals 
zero at the poles. 
54. For extreme values attainable for the declinational (D) factors, 
consideration must be given to the greatest declination which can 
be reached by the tide-producing body. The periodic maximum decli- 
nation reached by the moon in its 18.6 year node-cycle is 28.6° but 
this may be slightly increased by other inequalities in the moon’s 
motion. The maximum declination for the sun, taken the same as the 
obliquity of the ecliptic, is 23.45°. The declination factor of the 
long-period term (F',39 /g) has a maximum value of 2/3 when the decli- 
nation is zero. It diminishes with increasing north or south declina- 
tion but must always remain positive because of the limits of the 
declination. For the diurnal term (F,3, /g) the declinational factor 
has its greatest value when the declination is greatest. For the moon 
the maximum value of this factor is approximately 0.841 and for the 
sun 0.730. This factor is positive for the north declination and 
negative for the south declination. For the semidiurnal term (F532 /g) 
the declinational factor for both moon and sun is always positive and 
has a maximum value of unity at zero declination. 
55. The greatest numerical values for the several terms of the 
vertical component of the tide-producing force as represented by 
formula (32) and applicable to the time when the moon and sun are 
nearest the earth, are as follows: 
Greatest F',3) /g=—.070X 10-8 for moon, or —.027X10-$ for sun (33) 
Greatest F 3, /g= +.08810-* for moon, or +.03010-° for sun (34) 
Greatest F432 /g=+.105<10-§ for moon, or +.041X107° for sun (35) 
For the long-period term (33) the greatest value applies to either pole 
and is negative. For the diurnal term (34) the greatest value applies 
in latitude 45° and may be positive or negative according to whether 
the latitude and declinational factors have the same or opposite 
signs. For the semidiurnal term (35) the greatest value applies to 
the equator and is positive. 
56. Referring to formula (32), let a/c equal the mean value of 
parallax a/d. Then a/d may be replaced by its equivalent (a/c) (c/d), 
in which the fraction c/d expresses the relation between the true and, 
the mean parallax. Also let U=(M/E) (a/c)?, the numerical value 
of which will be found in table 1. Expressing separately the three 
terms of formula (32), we then have 
