HARMOISriC AISTALYSIS AND PREDICTION OF TIDES. 81 



A(n-i) COS (n— 1) p u = Ho cos {n—1) p u 



+ CjCOs (n— l)iicos (n—l)'pu+ 0.2 cos 2 (n— l)wcos {n—l)pu + 



+ Ck cos (n—l) Tc u cos (n—1) pu 



+ <S'iSm {n— 1) -ucos (n— l)23'M + 5'2sin2 {n—\)ucoQ {n— 1) fu-\- 



+ S\ sin {n—l) I u cos (n—l) p u (307) 



Summing the above equations 



a=(ii— 1) a=(n— 1) 



S ^a COS a p u = Ho S COS a 2> u 



a=o a=o 



a=(n.— 1) a=(n— 1) 



+ Cj S COS a ti COS a p u + Si 2 sin a -u cos a p u 



a=o a=o 



a=(n— 1) a=(n— 1) 



+ C'2 S COS 2a -u cos a p u + S^ S sin 2a w cos a p u 



a=o a=o 



a=(n-l) a=(B-l) 



4- Ck S cos a fc u cos a p u + Si S sin a Z ^i cos a p u 



a=o a=o 



a=(n— 1) m=k a=(n— 1) 



= So S cos a 2> '?^ + S C'm S cos a w w cos a p -u 



a=o m=l a=o 



m=l a=(n— 1) 



+ S 'Sm S sin a m tt cos a ^^ '^ (308) 



m=l a=o 



Examining the limits of (308), it will be noted by a reference to 

 page 77 that Ic, the maximum value of m for the C terms is ^ when n 



is even and ^ when n is odd; also, that I has a value of k— 1 when 



n— 1 

 n is even and —^ when n is odd. The limits of p, which is a partic- 

 ular value of m, will, of course, be the same as those of m. 



a=(n-l) 



By (294) the quantity S cos a p u becomes zero for all the 



a=o 



a=(n-l) 



values of p, and the quantity ^ cos a m u cos a p u becomes zero 



a=o 



for all values of m and p except when p equals m. By (294) , (299) , 



a=(n-l) 



and (302) the quantity 2!J sin a m u cos a p u becomes zero for all 



a=o' 



values of m and p. 



Formula (308) may therefore be reduced to the form 



a=(n-l) a=(n-l) 



S h^ COS a p u=Cf S cos^ a p u ^309) 



a=o a=o 



For any value of p less than ^ 



a=(n-l) 



S cos^ ap u=^\n (298) 



a=o 



