34 U. S. COAST AND GEODETIC SURVEY 
TERMS INVOLVING 4TH POWER OF MOON’S PARALLAX 
102. Formulas (24) and (26) represent the vertical and horizontal 
components of the part of the tide-producing force involving the 4th 
power of the moon’s parallax. This part of the force constitutes 
only about 2 percent of the total tide-producing force of the moon 
and for brevity will be called the lesser force to distinguish it from the 
principal or primary part involving the cube of the parallax. The 
vertical component F,,4 /g has its maximum value when 2 equals zero 
and, if numerical values pertaining to the moon and sun when nearest 
the earth are substituted in formula (24), the extreme values for this 
component are found to be 0.371078 for the moon and 0.385107" 
for the sun. The horizontal component F,,4 /g has its greatest value 
when 2 equals about 31.09° and the substitution of numerical values 
in formula (26) gives the extreme value of this component as 
0.26 107° for the moon or 0.24107" for the sun. 
103. Substituting in (24) the value of cos z from (31), the vertical 
component of the lesser force is expanded into four terms as follows: 
Fy, /g=15/4 (M/E) (a/d)* sin Y (cos? Y—2/5) sin D(5 cos? D—2)_  Fyao /g 
+45/8 (M/E) (a/d)*cos Y (cos? Y—4/5) cosD (5cos?D—4) cost Fy /g 
+45/4 (M/E) (a/d)* sin Y cos? Y sin D cos? D cos 2t______- Frye |g 
“hols (MLE) (a/d) cos icos Dicosst: eee eae eee Pug |g 
(132) 
These four terms represent, respectively, long-period, diurnal, semi- 
diurnal, and terdiurnal constituents, according to the multiple of the 
hour angle ¢t involved in the term. Each term is followed by a symbol 
wach is analogous to those used in the development of the principal 
orce. 
104. Each term in formula (132) may be further expanded by means 
of the relations given in formulas (39) and (42). Expressing these 
terms separately we hayve— 
Fy /g=15/4 (M/E) (a/d)* sin Y(cos? Y—2/5) X 
[3 (sin [—5/4 sin’ J) cos (7—90°) 
+5/4 sin? I cos (87—90°)] (133) 
Fy /g=45/8 (M/E) (a/d)* cos Y (cos? Y—4/5) X 
[5/4 sin? I cos? 41 cos (X—37) 
+ (1—10 sin? 4/+ 15 sin‘ 3/7) cos? 4 cos (X—7) 
+(1—10 cos? 47+ 15 cos* 4/) sin? 47 cos (X+7) 
+5/4 sin? I sin? 4J cos (X+3)7)] (134) 
Fy» /g=45/8 (M/E) (a/d)* sin Y cos? YX 
[sin J cos* 47 cos (2X—3j)+ 90°) 
+3 (cos? 4/—2/3) sin I cos? 41 cos (2X—7—90°) 
+3 (cos? 4J—1/3) sin I sin? 4] cos (2X+7—90°) 
+sin I sin‘ 47 cos (2X+37—90°)] (135) 
Fug [g=15/8 (M/E) (a/d)* cos? YX 
[cos® 37 cos (3X—3)) 
+3 cos 4/ sin? $I cos (3.X—7) 
+3 cos’ 4/ sint 4/7 cos (8X+7) 
+sin® 4/7 cos (3X+37)] (136) 
105. If the common factor (a/d)* in formulas (133) to (136) is 
replaced by its equivalent (a/c)* X (c/d)*, these formulas may be de- 
