HARMONIC ANALYSIS! AND PREDICTION OF TIDES 13 
relationship. The denominators of (11) and (12) are to be consid- 
ered as positive. 
41. Substituting in equations (8) and (9) the equivalents for b, sin 
CPR, and cos CPR from equations (10) to (12), the following basic 
formulas are obtained for the vertical and horizontal components of 
the tide-producing force at any point P at r distance from the center 
of the earth: 
me cos z—7/d 
Fr, (Sop i 1—2(r/d) cos 2+ (r/d)*}3 ays z| (13) 
M sin 2 : 
Felv=-3| aay cos eT GFT ~™ *| 
42. To express these forces in their relation to the mean accelera- 
tion of gravity on the earth’s surface, represented by the symbol g, 
we have 
glu=E/a*, or pu/g=a'/E (15) 
in which F is the mass and a is the mean radius of the earth. Sub- 
stituting the above in formulas (13) and (14), we may write 
cos z—r/d 
F, [9= (MIB) (ala)? | ppp Oe acon 2] (16) 
sin 2 
P, [g= (MIB) "| peg eae gem | 0 
43. Formulas (16) and (17) represent completely the vertical and 
horizontal components of the lunar tide-producing force at any point 
in the earth. If 7 is taken equal to the mean radius a, the formulas 
will involve the constant ratio M/E and two variable quantities— 
the angle z which is the moon’s zenith distance, and the ratio a/d 
which is the sine of the moon’s horizontal parallax in respect to the 
mean radius of the earth. Because of the smallness of the ratio a/d 
it may also be taken as the parallax itself expressed as a fraction of a 
radian. The parallax is largest when the moon is in perigee and at 
this time the tide-producing force will reach its greatest magnitude. 
A more rapid change in the tidal force at any point on the earth’s 
surface is caused by the continuous change in the zenith distance of 
the moon resulting from the earth’s rotation. The vertical com- 
ponent attains its maximum value when z equals zero, and the hori- 
zontal component has its maximum value when 2 is a little less than 
45°. Substituting numerical values in formulas (16) and (17) and 
in similar formulas for the tide-producing force of the sun, the fol- 
lowing are obtained as the approximate extreme component forces 
when the moon and sun are nearest the earth: 
Greatest F’, /g=.14410~° for moon, or .054 107° for sun (18) 
Greatest F, /g=.107 107° for moon, or .04110~° for sun (19) 
The horizontal component of the tide-producing force may be meas- 
ured by its deflection of the plumb line, the relation of this component 
to gravity as expressed by the above formula being the tangent of 
the angle of deflection. Under the most favorable conditions the 
