HARMONIC ANALYSIS. 



87 



m 



ao^-{yo.cosO+y,.cos3h + y,.cosQh + y3.cos9h + . . . +2/„-i.cos 3 [rn-l]/i) 

 m 



bs^^{yo-smO + yi.sm3h + y2.smQh + y3.sm9h+. . . +t/„-i.sin S[m-l]h) 



2^ 



m 

 hi=- (yo.sin + ?/i.sin ih + t/i.sin2i/i +y3.sin Zih+ . . . +2/„.i.sin i {m-\\h) 



a.= -(yo.coa0+yi.cosih + y2.cos2ih+y3.cos3ih+ . . . +?/„_i.cos i [m-l]h) 



' m 

 2 



m 



The principles of the sinusoid harmonic analysis may be deduced in 

 a somewhat different way * Starting from the theorem that any periodic 

 function can be expressed as the sum of a series of harmonic sinusoids 



?/=C + Ci.8in(/(;f— 9i)+C2. sin (2/0^ — 52) +C3. sin (3/ci — 53)+ ... (17) 



we have the problem of finding the values of the constants C, Ci, Ci, d, 

 . . . from the series of observed values xj^, xji, 2/2, Ih, ■ ■ ■ The equa- 

 tion (17) can be expanded in the form of a series of sines and cosines, thus, 



(13) 



y=C + ai.cos A;i + a2.cos 2A;i + a3.cos 3fci+ . . . 



+ 6,.siu A;f + 62.sin Ikt^-h-^m Zkt+ . . . 



where 



= — c.sing 6 — c.cos g, 



that is, (11) 



c = i/^M^' *^" 9^ -^• 



For the values of the ordinates we have 



?/o=C + a,.cos fco^ + aa.cos 2/co^+ . . . +&i.«in fcof + fe2.sin 2fcof+ . . . 

 r/j=C + ai.cos A;i^+ 02.003 2/cirH . . . +&i.sin A;i^ + Min 2/b,f + . . . 

 y,= C + ai.co9 A;2« + a2.co3 2kit-{- . . . +bi.sm kit + h.sm 2k;t+ . . . 



• • • * 



where ^'o^ ht, ht, ... are the arguments for the respective ordinates. 

 When the m ordinates are at equidistant intervals h throughout the whole 



♦Bessel, Ueber die Bestimmung des Gesetzes einer periodischen Erscheinung. Astronom. Nachr., 

 VI, 333; Abhandlungen, ii, 364, Leipzig, 1876. 



