where d is the mean distance, and d' the deviation from the mean, the binomial theorem 

 can be used to convert the variable part of the denominator in equation (15) into a power 

 series in d'/d. Thus, 



d"3 = d" 



-(f)-(fj- 

 d-4 = d-4 Ti - 4 (4') + 10 (4'Y - 



(17) 



The term d/d is a quasi-periodic function, whose ampUtude for the Earth-Moon system 

 varies from approximately 0.04 to 0.07 with an average value of 0.055. For the Earth-Moon 

 system, the amplitude of this term is about 0.017. Thus, only tlae first two terms of 

 equation (17) need to be considered. The period of the variation in distance is the 

 anomaUstic year of 365.259 solar days for the Sun and the anomalistic month of 27.554550 

 solar days for the Moon. The slightly higher tidal ranges when the Moon is nearest the Earth 

 are called perigean tides. The sUghtly lower tidal ranges when the Moon is farthest from the 

 Earth are called apogean tides. The effect of tlie variable distance from Earth to Sun is much 

 smaller and does not often receive any special recognition. Trignometric identities can be 

 used to combine the trignometric expressions arising from the variabihty of distance with 

 the trignometric functions which depend on the angle z. 



The angle z is obviously a function of the latitude of P, and the hour angle which 

 results from the Earth's rotation beneath the Sun and Moon. It is also a function of the 

 declination of the Sun and Moon. The importance of these latter factors is further explained 

 below. 

 2. The Declination of the Sun and Moon. 



It is weU known that the equatorial plane of Earth is inclined to the plane of the 

 revolution of the Earth around the Sun by an angle of 23° 27'. As a result, the Sun appears 

 to be directly above the Equator at the vernal equinox, 23° 27' north of the Equator at tlie 

 summer solstice, above the Equator again at the autumnal equinox, and 23° 27' south of the 

 Equator at the winter solstice. Figure 9 shows that the high point of the tidal bulge due to 

 the Sun will be north of the Equator in the Northern Hemisphere by day and south of the 

 Equator by night near the summer solstice. The tidal bulge due to the Sun is centered over 

 the Equator at the equinoxes. Thus, the contribution of the Sun to the two daily tidal 

 bulges will be nearly equal in both hemispheres near the times of the equinoxes when the 

 Sun is above the Equator. When the Sun is near the northern or soutliern limits of its orbit 

 as seen from tlie Earth, a fixed point on the tidal bulges which it produces in each 

 hemisphere will be unequal. 



30 



