82 U. S. COAST AND GEODETIC SURVEY. 



but when V~'^i ^^ quantity becomes equal to n (301). 

 Therefore for all values of p less than ^ 



2 a=(n-l) 



Cp=- S Tiacosa'p u (310) 



n a=o 



but when p is exactly ^ 



I a=(n-l) 



Cp = — S hg, COS a p u (311) 



^ a=o 



Since in tidal work p is always taken less than k, we are not espe- 

 cially concerned with the latter formula. 



To obtain the value of any coefficient S, such as S^, multiply each 

 equation of (303) by sin' a p u. Sum the resulting equations and 

 obtain 



a=(n— 1) a=(n— 1) 



, S ^a sin a p u = Ho S sin a p u 



a=«o a=o 



in=k a=(n— 1) 



+ S ^m S COS a m u sin a pw 



in=l a=o 



m=l a=(ii— 1) 



+ S 'S'm S sin a m li sin a p w (312) 



ia=l a=o 



a^(ii-l) 



By (294), (299), and (302) the quantities S sin a p u and 



a=Gi 

 a = (n-l) 



S cos a m u sin a p u are zero for all the values of m and p; 



a=o 



a=(ii-l) 



and S sin a m u sin a p u becomes zero for all the values of m 



a=o . > 



and p except when m and p are equal. In this case the limit of I fca" 



^ ^ a-(n-l) 



m and p is less than - and by (297) , the quantity S sin^' ai p w 



= \ n. 



Therefore, formula (312) reduces to the form 



a=(n-l) 



S hasm a p u==^ n S^ (313) 



a=o 



and 



2 a=(n-l) 



8-p = - S ^tL sin a p w (314) 



Tl a=o 



