HARMONIC ANALYSIS AND PREDICTION OF TIDES 63 
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=H+C, cos 0+ OC, cos 26+ -_-_________- CO, cos k 6 
+ §, sin 6+ S, sin 20+ _____ Le S, sin l 6 
m=k m=/ 
=HM)+ >)5 Cn cos m6+ >) Sp sin mé (257) 
m=1 m=1 
; , See i. ‘ n—1. : 
in which the limit k=5 if n is an even number, or k= 5 if n is an 
odd number; and the limit l=5-1 if m is even, Of ae Dat m 1s odd, 
190. By substituting successively the coordinates of the n given 
points in (257) we may obtain n equations of the form 
m=k m=! 
ha=Ho+ >) Cn cos mau+ >) S,, sin mau (258) 
m=1 m=1 
in which a represents successively the integers 0 to (n—1). 
By the solution of these n equations the values of n unknown 
quantities may be obtained, including H, and the (n—1) values for 
Cy and Sy. It will be noted that the sum of the limits k and 1 of 
(257) or (258) equals (n—1) for both even and odd values of n. 
191. The reason for these limits is as follows: 
A continued series 2 C,, cos m a u may be written 
C, cosautC, cos2au+___.+C, cosnau 
+ Cas cos (n+1) aUutCas., cos (n+2) au+___._+C,,cos2nau 
+ Cent) COS (2n+1) @ U+C en+12, Cos (2n-+2) a U+ ___- 
+03, cos3 NAU 
Ee See NN eR pgm le oa! ah aye RIAL. ett (259) 
Since n w=27 and a is an integer, the above may be written 
[C+ City + Cent + eassescs | cosau 
aes, Con42) + Cent2) + sececos< J cos2au 
(Cee Oe eCae tc. Jeos(n—1)au 
+[(C,+Cy+Cm+-------- |cosnau (260) 
Since cos n a u=cos 2a r=1; cos (n—1) au=cos (2a r—a4 U)=COS AU; 
cos (n—2) a u=cos 2 a u; ete., (260) mav be written 
(C+ Con + Con I | COS 0 
(Ona Cia Cepan ans aoe: =e 
Sr Olin OL Ca Opel OCC i SS55 S555 ] cos au 
a [C; = Cnt) at Cons) =P BSeSSs= os 
+ Cn —2) + C en-2) + Cian—2) + ee |cos2au 
Hl Creer Cipanyar Cease as = 222s 
Ca-1 + Cony + Cen-1 + --e- eee ] cos k au (261) 
The first term of the above is a constant which will be included with 
the H, in the solution of (258). From an examination of (261) it is 
evident ee the cosine terms will be completely represented when 
== 
— oy one a? according to whether 7 is even or odd. 
ee the continued series = S,, sin m a u may be written 
