IIAEMONIC ANALYSIS AND PEEDICTION OF TIDES. 



79. 



If 2> and m are equal integers and do not exceed kj formulas (291) , 



sin (p — m)?: 

 (292), and (293) will contain the indeterminate quantity . p — m 



sin ^ 71 



n 







n 



= pr, and also when p and m each equal k, the indeterminate quantity 

 sin {p-^m)7i _0 



. ip + m) 

 sin ~ -TT 



Evaluating these quantities we have 



sin (p — m)7c' 



sin 



p — m 



71 cos {p — m)7z' 



IT p — m 



, ^ ~ cos TZ 



ip — mj^o n n 



{p — m)=o 



n (295) 



and 



sin {p + 171)71' 



. p + m 



sin 71 



n 



TT cos ip + m)n' 



n p-\-m = -71 (296) 



/- , \ ~ cos TT . , . 



(p + 7?i) = n n- n J (p + m) = ?i 



In (296) it will be noted that when the integers p and m each equal 



^, n must be an even number, and therefore cos mi is positive, while 



cos 71 is negative. 



Assuming the condition that p and m are equal integers, each less 



than K, we have by substituting (295) in (291), (292), and (293), 



a=(ii— 1) a=(ii— 1) 



S sin a p u sin a m u= S sin^ a m u = ^ n 



a=o a=o 



i=(n-l) a=(n— 1) 



S COS a p u cos a mu= S cos^ a mu = ^ n 



a=o a=o 



(297) 

 (298) 



a=(ii— 1) _ a=(n— 1) 



S sin a 2> li cos a m u= S sin a m u cos a m li = (299) 



a=o a=o 



Assuming the condition that p and m are each equal to ^, we have 

 by substituting (295) and (296) in (291), (292), and (293), 



a=(ii-l) 



S sin^ amu = ^n + ^n cos 7r = 



a=o 

 a=(n-l) 



S cos^ a mu = ^ n — ^ n cos 7t = n 



a=o 



a=(n-l) 



S sin a mu cos a m u = 



a=o 



(300) 

 (301) 



(302) 



