38 U. S. COAST AND GEODETIC SURVEY 
argument. With this change and allowing for differences in the 
latitude factors and numerical coefficients, the summarized formulas 
for the west component will be similar to those for the vertical com- 
ponent and may be written as follows: 
Fy, /g=1/2 U (cos?¥—4/5) 2 fC cos (E—90°) (168) 
Fy. /g=1/2 U sin 2Y 2 fC cos (H—90°) (169) 
Fy [g=3/2 U cos*Y 2 fC cos (E—90°) (170) 
114. To obtain the horizontal component of the lesser force in any 
direction, the same procedure may be followed as was used for the 
principal tide-producing force (paragraphs 85 to 87). With the same 
system of notation we then have 
Fy /g=3/2 U cos Y (cos? ¥ —4/5) cos A 2 fC cos (H+180°) (171) 
Foy ig—3/2 U0 Py Df cos (=e) (172) 
Fug |g=3/2 U P, = fC cos (H—X,) Gis) 
Fyu3 /g=3/2 U P3 2 fC cos (H—X3) (174) 
in which 
P,=(sin’Y (cos? ¥Y —4/15)? cos?A+1/9(cos? Y —4/5)? sin?A]* (175) 
P.=cos Y|(cos?¥Y —2/3)? cos?A+4/9 sin?Y sin?A]” (176) 
P;=cos’Y (sin?Y cos?A-+sin?A)” (177) 
renk eh (cos? Y —4/5) sin A 
AG — tan 3 sin Y(cos?Y—4/15) cos A (178) 
s 2 simi sin Al 
oe en 23 (cae = 2/3) cos A a) 
XG tana ee (180) 
sin Y cos A 
The proper quadrants for Xj, X52, and X3 will be determined by the 
signs of the numerators and denominators in the above expressions, 
these signs being respectively the same as for the sine and cosine of 
the corresponding angles. 
115. Comparing formula (122) for the equilibrium height of the 
tide due to the lesser tide-producing force with formula (24) for the 
vertical component of the force, it will be noted that they are the 
same with the exception that the numerical coefficient of the former 
is one-third that of the latter. With this change, the summarized 
formulas (151) to (154) for the vertical force may be used to express 
the corresponding equilibrium heights. Following the same system 
of notation as before, we have 
hy /a=1/2 U sin Y(cos?¥Y—2/5) = fC cos E (181) 
hy /a=1/2 U cos Y(cos*?Y—4/5) = fC cos E (182) 
hy /a=1/2 U sin Y cos?Y 2 fC cos EF (183) 
lug [a=1/2 U cos’¥ 2 fC cos E (184) 
It is to be noted that the equilibrium height of the tide due to the 
principal tide-producing force when measured by the mean radius of 
the earth as a unit is one-half as great as the corresponding vertical 
component force referred to the mean acceleration of gravity as a 
unit, while the equilibrium height due to the lesser tide producing 
force similarly expressed is only one-third as great as the corresponding 
force. In table 2, the coefficients (C) of the constituents derived 
