76 U. S. COAST AND GEODETIC SURVEY. 



possible to find values for Ho, Cm, and S^, which when substituted in 

 (276) will give the equation of a curve that will pass exactly through 

 each of the 24 points representing the component means. 



In order to make the following discussion more general, let it be 

 assumied that the period of Q has been divided into n equal parts, and 

 that the ordinate or value of h pertaining to the beginning of each of 

 those parts is known. Let u equal the interval between these ordi- 

 nates, then 



ntt = 27r, or 360° (277) 



Let the given ordinates be h^, h^jTi^- ■ • • h {n-i) corresponding to 

 the abscissae o, u, 2u • • - ■ (n—1) u, respectively. 



It is now proposed to show that the curve represented by the 

 following Fourier series will pass through the n points of which the 

 ordinates are given. 



h = Ho+C^cosd+C2COs2d+ . . . . Cy, cos k d 

 + Sismd + S2sm2d+ . . . . Si sin I 6 



iii=k m=l 



= -ffo + S Cm cos m. 6 + ^ Sjn sin md 



in=l m=l 



(278) 



in which the limit A: = ^ if ri is an even number, or fc = ^ if n is an 



odd number; and the limit l = ^—\ ifTiis even, or ^ if n is odd. 



By substituting successively the coordinates of the n given points 

 in (278) we may obtain n equations of the form 



m=k ni=l 



i^a = -ffo+ S Cm cos mau + S Sra siu mau (279) 



m=l m=l 



in which a represents successively the integers to (n— 1). 



By the solution of these n equations the values of n unknown 

 quantities maybe obtained, including Ho and the (n— 1) values for 

 Cra and Sm.- It will be noted that the sum of the limits Ic and I of 

 (278) or (279) equals (n— 1) for both even and odd values of r? 

 The reason for these limits is as follows : 

 A continued series 2 Cm cos mau may be written 



C^ cos au-\- C^ cos 2 a u + • • • • + C^ cos n a u 



+ C(n+i) cos {n -r 1) au+ (7(n+2) cos {n + 2) au-V ■ • • • + Cjn cos 2na l 

 + C (211+1) cos (2n+ 1) a u+ (7(211+2) cos (271 + 2) a u+ • • • • 

 + C'gn cos Sn a u 

 + (280) 



Since n u =2ti and a is an integer, the above ma} be written 



[6'i+(7(n+i)+C(2n+i)+ • • • ']cosau 

 + [6',+ 6'(n+,)+ C(2i,+2)+ • • • • ] COS 2 a w 

 + 



+ [C(n_i)+C(2n-i)+C*(3n-i)+ • • • • ] COS (?^-l) au 



+ [Cn+C2n+<^'3n+ • • • • ] COS n « u ■ (281) 



