HAEMONIC AJSTALYSIS AND PEEDICTIOI^ OF TIDES. 77 



Since cos n a u = cos 2a tt = 1 ; cos (n-l) a u = cos (2a w—au) = cos a v ; 

 cos {n—2) a ^t = cos 2 a tt; etc., (281) may be written 



[Cn+C2n+C'3n+ • • • • ] COS 

 + [Ci+ C(ii + i)+ C(2n+i)+ • • • • 

 + C(ii_i)+ C(2n-i)+ ^(3n-i)+ • • • • ] COS a U 

 + [C2+ C(n+2)+ C(2n+2)+ • • • • 



+ C(n-2)+ ^212-2)+ ^(3n-2)+ • • • • ] COS 2 a u 

 + [Ck+ C(n+k)+ C(2n+k)+ • • • • 



+ C(n-k)+ C'(2n-k)+ C'(2j3^_k)+_j • • • ] COS fc a M (282) 



The first term of the above is a constant which will be included with 

 the Ho in the solution of (279). From an examination of (282) it is 

 evident that the cosine terms will be completely represented when 



^ = K, or — ^ — , according to whether n is even or odd. 



Similarly, the continued series 2] ^m sin m a -u may be written 

 [/Sn + *S'2n + 5'3n+ . . . . ] sin 



+ [5'i + /S'(n+i) + «S'(2n+i)+ .... 



— 5'(n_i) — /Sczn-i) — 'S'(3n — ^) _ . . . . ] &m. a U 

 + [S'2 + 5'(n+2) + 'S'(2n+2) + • • . • 



— *S'(n_2) — iS'(2n-2) — «S'(3n-2)— • . . . ] siu 2 a 'U 



+ [/Si + 5'(n+l) + ^(2n+l) + .... 



— 'S'(n_i) — <5(2n-i) — ^(3n-i) — • • • • ^suilau (283) 



The first term in the above equals zero. The remaining terms will 



take complete account of the series S'S'm sin m a w, if Z = ^c — 1 when 

 n is even, or „ when n is odd. 



From the foregoing it is evident that the limit of mwill not exceed ^' 



If we let u and a represent any angles with fixed values, m and p 

 any integers with fixed values, and a an integer having successive 

 values from to (n— 1), it may be shown that 



a=(n--l) _ sin^ n m W • ri / 1^ i i fncA\ 



S sm (a m It + a) = -^-^-j sm[^ {n-1) m u + a] (284) 



a=o sm t m u 



S cos (a m u + a) = — ^^S cos[^ [n-l) mu + a] (285) 



a=(n-i) _ _ sin ^ n ip — m) u COS ^ (n—l) (p — m) u 



S Binapusmamu:^^ sin Up-m)u " 



_ sin i n ip + m) u cos i (n- 1) jp + m) u .^gg) 



* sin i ip + m) u 



a=(n-i) sin ^ n ip — m) u cos ^{n —1) {p — m)u 



S cos a p -M cos a m it = ^ ■ — r~r S ■ ' 



^0 ^ sm ^ {p-m) u 



sin ^ n ip + m) u cos j (n— 1) (p + m) u (2^^^ 



^ sin i {p-\-m) u 



^^(p-i) . sin I 7^ ip — m) u sin ^ (n—l) {p — m) u 



S sin a p u cos a m u = h = — n ^; 



a=o sm t (p — ^) '?^ 



^ sin -^ 71 (p + m) ^ sin ^ (n-1) jp + m) u , . 



^ sin ^ (p + m) ii 



