24 U. S. COAST AND GEODETIC SURVEY 
74. Coefficients—-The complete coefficient of each term of formulas 
(62) to (64) includes several important factors. First, the basic factor 
U, which equals the ratio of the mass of the moon to that of the earth 
multiplied by the cube of the mean parallax of the moon, is common 
to all of the terms. This together with the common numerical coeffi- 
cient may be designated as the general coefficient. Next, the function 
involving the latitude Y is known as the /atitude factor, each formula 
having a different latitude factor. Following the latitude factor is a 
function of J, the inclination of the moon’s orbit to the plane of the 
earth’s equator, which may appropriately be called the obliquity factor, 
each factor applying to a group of terms. Lastly, we have an indi- 
vidual term. coefficient which includes a numerical factor and involves 
the quantity e or m. Since these factors are derived from the equa- 
tions of elliptic motion, they will here be referred to as elliptic factors. 
The product of the elliptic factor by the mean value of the obliquity 
factor is known as the mean constituent coefficient (C). Numerical 
values for these coefficients are given in table 2. Since all terms in 
any one of the formulas have the same general coefficient and latitude 
factor, their relative magnitudes will be proportional to their constitu- 
ent coefficients. Terms of different formulas, however, have different 
latitude factors and their constituent coefficients are not directly 
comparable without taking into account the latitude of the place of 
observation. 
75. The obliquity factors are subject to variations throughout an 
18.6-year cycle because of the revolution of the moon’s node. Dur- 
ing this period the value of J varies between the limits of w—7 and 
w+i, or from 18.3° to 28.6° approximately, and the functions of J 
change accordingly. In order that tidal data pertaining to different 
years may be made comparable, it is necessary to adopt certain stand- 
ard mean values for the obliquity factors to which results for different 
years may be reduced. While there are several systems of means 
which would serve equally well as standard values, the system adopted 
by Darwin in the early development of the harmonic analysis of tides 
has the sanction of long usage and is therefore followed. By the 
Darwin method, the mean for the obliquity factor is obtained from 
the product of the obliquity factor and the cosine of the elements £ 
and y appearing in the argument. This may be expressed as the 
mean value of the product J cos u, in which J is the function of J in 
the coefficient and wu the function of € and v in the argument. Since 
u is relatively small and its cosine differs little from unity, the result- 
ing mean will not differ greatly from the mean of J alone or from the 
function of J when given its mean value. 
76. Using Darwin’s system as described in section 6 of his paper 
on the Harmonic Analysis of Tidal Observations published in volume I 
of his collection of Scientific Papers (also in Report of the British 
Association for the Advancement of Science in 1883), the following 
mean values are obtained for the obliquity factors in formulas (62) to 
(64). These values were used in the computation of the corresponding 
constituent coefficients in table 2. The subscript , is here used to 
indicate the mean value of the function. 
For terms A, to A; in formula (62) 
[2/3—sin? I= (2/3—sin? w) (1—3/2 sin? 1) =0.5021 (65) 
For terms A, to Aj; in formula (62) 
[sin? I cos 2é])>=sin? w cos! 44=0.1578 (66) 
