80 



U. S. COAST AND GEODETIC SUE^^EY. 



Returning now to the solution of (279), by substituting the suc- 

 cessive values of a from to (n — 1) , we have 



ho = Ho+ Ci cos + C, cos + .... + Ck cos 

 + 5i sin + Sl sin 0+ ... . +Si sin 



hi = Ho+ C^ cos u+ C^ cos 2u+ .... + Cb cos Tcu 

 + Sy^ sin u + S^Qyn. 2u+ . . . . +Si sin lu 



h2 = Ho+ Ci cos 2u + Cj cos 4:u+ .... + C^ cos 2lcu 



+ 5, sin 2^ + S'2 sin 4?^+ .... +^i sin 2^1^ y (303) 



^(n_i) = 5'o+ Cj cos (n— l)ti+ 6*2 cos 2(n— 1)^/,+ .... 

 + Ck COS (n — l)Z:^t 

 + 8^ sin {n—l)u + S2 si'D.2{n~l)u+ .... 

 + (§1 sin (n— l)lu 



To obtain value of Ho, add above equations 



a=(n-l) 



S h^ = n Ho 



a=o 



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



-1- Ci S COS a w + Cg S COS 2 a u+ . . 



a=o a=o 



a=(n-l) 



+ Cb S COS a fc w 



a=o 



a=(ii-l) _ 



+ Si S sin a Z w 



a=o 



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



-fS'i S sin a u + S^ S sin2au + 



a=o a=o 



m=k a=(n— 1) m=l a=(n— 1) 



= n^o+SC'm S cosamu+S'S'm S sin a. m w (304) 



in=l a=o m=l a=o 



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



From (294) , S cos a m u and S sin a m ^i each equals zero, 



a=o a=o 



7} 



since neither fc nor I, the maximum values of m exceeds ^ 

 Therefore 



and 



S n^ = n Ho 



a=o 



(305) 



1 a=(ii-l) 

 ii' a=o 



(306) 



To obtain the value of any coefficient C, such as (7p, multiply 

 each equation of (303) by cos a p u. Then 



^0 cos = Ho cos 



+ (7i cos + C, cos + . . . . + Ck cos 

 . +Si sin + ^2" sin 0+ ... . +Si sin 



h^ cos p u = Ho cos p u 



+ Ci cos u cos pu+ C2 COS 2ii cos p t^ + . . 

 + /Si sin u cos p U + S2 sin 2u cos p u+ . . 



7i2 cos 2p u = Ho cos 223 u 



+ Ci cos 2u cos 2p u + C2 cos 4w cos 2p u + 



+ Ck cos 2fc tt cos 22? u 

 + /Si sin 2u cos 22? u + /S2 sin 4w cos 22? w + 



+ 5'i sin 2l u cos 22? u 



+ Ck cos Z: u cos 2> tt 

 + S\ sin Z u cos p u 



