The Tide-generating Forces 265 



where U is not constant, but varies around an average value u with the 

 frequency a', so that 



U = u+u'cos(a't-x'); (VIII. 15) 



generally a' is much smaller than a (e.g. if U contains the fortnightly variation 

 of declination, the period in a' is approximately 14 days for the moon, and 

 6 months for the sun, whereas the period in a is \ day for the moon and 

 1 day for the sun). Substituting in (VIII. 14), we obtain 



Q 1 = wcos(otf— *)+£ «' COS [(ff+ #')*—(* + «')] + 



+ Wco&[(o-o r )t-(x-x')] . (VIII. 16) 



Besides the variation with the frequency a, which is to be considered as the 

 fundamental one, there are two other oscillations with the amplitude \u' 

 which depends upon the variations of U, and with frequencies equal to the 

 sum and difference of the two frequencies. The same can be done with two 

 or more terms. The variables which determine the true motions of the sun 

 and the moon, and which follow from such a development in sine and cosine 

 terms are the following. 



If 6 is the sidereal time (hour angle of the vernal equinox), the hour angle 

 of the sun is 6 — h 1 , if h x is the real longitude of the sun eastward along the 

 ecliptic. Whereas increases practically uniformly, h x increases irregularly 

 in easterly direction. The sun is replaced by a fictitious sun supposed to move 

 steadily around the ecliptic at a rate which is the average rate of the true 

 sun and its longitude (h) is said to be the mean longitude of the sun, which 

 increases also steadily. Then 



t = e-h 



is the mean solar time. 



The same procedure is followed for the moon; if s is the mean longitude 

 of the moon, then 



t = e-s 



is the mean lunar time. 



From these relations follows: 



t = t-rh — s , where D = s — h , 

 is the angular difference between the mean moon and the mean sun. 



The other quantities appearing in Q, like distance, declination, real hour 

 angle, etc., can all be expressed by h and s and three other angles increasing 

 uniformly. These are: p, the mean longitude of the lunar perigee, p s , the mean 

 longitude of the solar perigee, and N' = —N, in which N is the mean longitude 

 of the ascending lunar node. These variables increase during a mean solar 

 day by the following amounts, whereas the length of their period (time of 

 a full revolution of 360°) is indicated beside: 



t 360°-12 91° 24 h 50 47 min (lunar day) p 01114° 8 847 years 



s 13176° 27-32 days (sidereal month) N 0529° 18 61 years 



h 0-986° 1 year p s 0000047° 20940 years 



