HARMONIC ANALYSIS! AND PREDICTION OF TIDES 15 
comes zero when z=cos-!+-/1/3 (approx. 54.74° and 125.26°). The 
horizontal component (25) has its maximum value when z=45° and 
an equal maximum negative value when z=135°. The horizontal 
component becomes zero when z=0, 90°, or 180. 
48. If numerical values applicable to the mean parallax of the moon 
are substituted in (23) and (25), these component forces may be 
written 
F'., /g at mean parallax=0.000,000,167 (cos? z—1/3) (27) 
F.3 /g at mean parallax=0.000,000,084 sin 22 (28) 
For the corresponding components of the solar tide-producing force, 
the numerical coefficients will be 0.46 times as great as those in the 
above formulas. 
49. For the extreme values of the components represented by (23) 
and (25), with the moon and sun nearest the earth, the following may 
be obtained by suitable substitutions: 
Greatest F’,3 /g=.140<10~* for moon, or .05410~° for sun (29) 
Greatest F,3 /g=.10510~* for moon, or .04110~° for sun (30) 
Comparing the above with (18) and (19), it will be noted that the 
maximum values of the lunar components involving the cube of the 
moon’s parallax are only slightly less than the corresponding maxi- 
mum values for the entire lunar force, while for the solar components 
the differences are too small to be shown with the number of decimal 
places used. 
VERTICAL COMPONENT OF FORCE 
50. It is now proposed to expand into a series of harmonic terms 
formula (23) which represents the principal vertical component 
of the lunar tide-producing force. In figure 3 let O represent the 
ee of the earth and let projections on the celestial sphere be as 
follows: 
C, the north pole 
I M’ P’, the earth’s equator 
I M, the moon’s orbit 
M, the position of the moon 
IP the place of observation 
CM M’, the hour circle of the moon 
CP P’, the meridian of place of observation 
I, the intersection of moon’s orbit and equator 
Also let 
I =angle M J M’=inclination of moon’s orbit to earth’s equator 
t =are P’ M’ or angle PCM=hour angle of moon 
X=IP’=\longitude of P measured in celestial equator from 
intersection J 
j =1M=longitude of moon in orbit reckoned from intersection J 
z =PM=zenith distance of moon 
D=M’M=declination of moon 
ee ett cen ee a 
