268 



little in a half tropical month. Designating the length of this re- 

 sultant at the middle of the month as K' , the mean value of Di 

 during the month is, very nearly : 



Dm'==C{K'-\-Oij{0)) (23A) 



in which C has the same numerical value as in equation (22 A). 



The correction to be added to the value of Ki for the month, to 

 give the value of K' , is to be derived. 



17. Correction jor Pi. — As shown in paragraph 122 the equation of 

 the Ki equilibrium component is : 



7/i = Ki cos (r+/i-90°-/) 

 and, from equation (69) that of the Pi component is 



7/2=Pi COS (r-/t+90°). 

 The angle between them is : 



)8' = 2/i-180°-/. 



in which h is the mean longitude of the sun (par. 105). Its value on 

 any given day of the year is substantially the same from year to 

 year. It increases at the rate of 0.041° per solar hour, or about 1° 

 per day. v' is a small angle, which varies with N, the longitude of 

 the moon's node. Its values corresponding to values of N are tabu- 

 lated in manuals on the harmonic analysis of tides. The value of /S' 

 on any date may be corrected for v' by taking the value of h on half 

 as many days before the given date as there are degrees in v' when 

 v' is positive and after the given date when v' is negative. When 

 so corrected the value of ^' is 



/3' = 2A,-180° 



The length, K' , of the resultant of the Ki and Pi components is 

 easily shown to be 



i^'^VK?+P?-2KiPi cos ^' 

 = VK?+P?-2KiPi cos 2h 

 Placing K'^cKi 



c=i^7Ki = Vl+P?/K?-2(Pi/Ki) cos 2fi (24A) 



Taking the value of Pi/Ki as the ratio of the mean values of the 

 coefficients of the corresponding equilibrium components, or as 

 0.0880/0.2655 = 0.3315, equation (22A) becomes: 



c=Vl-ll-0.663 cos 2h (25A) 



