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HARMOXIC ANALYSIS AISTD PEEDICTIOISr OF TIDES, 99 



Comparing (390) and (391), it will be found that 



A" cos «! = mean value of [J. cos a + S J5 cos { (5 - a) ^ + /3 }] (392) 



A^ sin Q:i=mean value of [A sin a + S 5 sin {(6-a) ^ + |S}] (393) 



Let T = length of series in mean solar hours. Then the mean value 

 of B cos{(6 — a) H-jS} within the limits ^ = and t = T, is 



5 cos {Q)-a)t + ^]dt = ^ (h-a)T ^^^^ {(fi-a)r + ^} -sin 0] 



180 sin 4(& — a)r „ ,,,-, , ^, ,r.^.^ 



-^ Uh-a)J Bcos{i{h-a)r + p} (394) 



The mean value of B sin{(6— a) M-jS} within the same limits is 



^-Tb sin {{h-a)t + ^}dt = ^ .^ J^ . ^ [cos{(6-a) r + iS} -cos ^] 



180 sin+ (6 — a)r „ . ,,,, , ^, /^^^^n 



= fTTT — v-^ B sm Uih-a)T + ^\ (395) 



Substituting (394) and (395) in (392) and (393), and for brevity 

 letting 



180 sin Mi- a) r ^ (306) 



we have 



A^ cos a'=A cos a + i: F^ cos {^{1-0)7 + ^} (397) 



A'^ sin ai=^ sin a + S Fb sin {i(&-a)T + |S} (398) 



Transposing, 



A cos a = A^ cos a^-X F^ cos {i(b-a)T + 13} (399) 



^ sin a = A^ sin a^-S Fb sin {i (h-a) t + /3} (400) 



Multiplying (399) and (400) by sin a^ and cos a^, respectively, 



A sin a^ cos a = A^ sin a^ cos a^ — 2 F^ cos {|(&— a)r + i3} sin a^ (401) 



A cos q:^ sin oc = A^ sin a- cos o;^ — S Fb sin {^(6 — a)T + ^} cos a^ (402) 



Subtracting (402) from (401) 



A sin (Q;i-a)=S Fb sin {i(&-a)r-!-^-a^} (403) 



Multiplying (399) and (400) by cos a^ and sin a^, respectively, 

 A cos a^ cos a = A^ cos^ a^-S Fb cos {\{h--a)T-\-^] cos a^ (404) 

 A sin q;1 sin a =A^ sin^ o;i-S Fb sin {i-(&-a)T-l-|3} sin a^ (405) 

 Taking the sum of (404) and (405) 



^ cos («!-«)= Jl1-S Fb cos {i(6-a)r-h/3-ai} (406) 



Dividing (403) by (406) 



tan U a) ^ nsin{i(6-a)r + /3-^M 

 tan Ka «;-^x_2 /^^ eos {i(6-a)r-r|S-«^} 



(407) 



