16 U. S. COAST AND GEODETIC SURVEY 
The solution of a number of the spherical triangles represented in 
figure 3 will provide certain relations needed in the development of 
the formulas for the tide-producing force. 
51. In spherical triangle MCP, the angle C equals ¢ and the sides 
MC and PC are the complements of D and Y, respectively. We may 
therefore write 
cos z=sin Y sin D+ cos Y cos D cos t (31) 
Substituting this value in formula (23), we obtain 
Fy /g=3/2 (M/E) (a/d)?(1/2—3/2 sin? Y)(2/3—2 sin’? D)___- Fis /g 
+3/2 (M/E) (a/d)? sin 2Y sin 2D cos t____ Fy 31 /g 
+3/2 (M/E) (a/d)’ cos? Y cos? D cos 2t____Fy39 | g (32) 
Cc 
FIGURE 3. 
52. In formula (32) the vertical component of the tide-producing 
force has been separated into three parts. The first term is inde- 
pendent of the rotation of the earth but is subject to variations aris- 
ing from changes in declination and distance of the moon. It in- 
cludes what are known as the long-period constituents, that is to say, 
constituents with periods somewhat longer than a day and in general 
a half month or longer. The second term involves the cosine of the 
hour angle (¢) of the moon and this includes the diurnal constituents 
with periods approximating the lunar day. The last term involves 
the cosine of twice the hour angle of the moon and includes the 
semidiurnal constituents with periods approximating the half lunar 
day. The grouping of the tidal constituents according to their 
approximate periods affords an important classification in the further 
development of the tidal forces and these groups will be called classes 
or species. Symbols pertaining to a particular species are often 
identified by a subscript indicating the number of periods in a day, 
