78 



U. S. COAST AND GEODETIC SURVEY. 



In 

 If we let q; = and u = — , or nu = 2Ti, then formulas (284) to (288) 

 n 



may be written as follows : 



/ m \ 



( m -K 71 ) 



a=(n-i) , smm7rsm^^m7r--7r 



S sin a m u = _ 



^ ^ sm — ;r 



n 



a=(n-l) sm m 7Z COSi m TT - - TT J 



S cosam'U = 



a=o 



(289) 



(290) 



. m 

 sm — It 

 n 



a=(n-i) sin (p-m) 7r cos (p-m) ^r-^-^^ tt 

 S sin a p w sin a m li = i — i'p-m) ' 



sin 



n 



7r 



sm 



-* 



/ N F/ N ?) + m "I 



yp — m) 71 cos (2? + m) tt — ^*^ tz 



. v + m 



sm *^ 7l 



n 



a=(n-i) sin (p-m) TT cos I (2?-m) ^^-^^^"^ H 



(291) 



S cos a p u cos a m u = i 



sin ^^ TT 



sin (p + m) Tz cos (p + w) tt — ^^- n 



+ i tr^:! (292) 



. v + m 



sm *^ 71 



n 



a=(ii-l) 



S sm a p u cos a m w = ^ 



sin ip — m) 7z sin {p — m)7z — tt 



p— m n 



. ?) — m 



sin 7t 



n 



+ h 



sin {p + m) 71 sin (p + m) tt — 7z 



. p + m 



sm TT 



m 



(293) 



If p and m are unequal integers and neither exceeds x, the above 

 (289) to (293) become equal to zero. Thus, 



(294) 



