HARMONIC ANALYSIS. 95 



If, for illustration, we have a = 10, then Ci=1.98, c, = 2.70, 

 C3 = G.85, C4 = 5.94, Cs = 1.75, c, = 0.94, c-, = 0.61, Cs =0.43, c^ = 0.32, 

 Co = 0.24, cn = 0.20, C12 = 0.16, c,3 = 0.14, c,4 = 0.12, Ci5=0.10, Ci6= 0.09, 

 Ci, = 0.08, c,8 = 0.07, Ci9 = 0.06, Cjo = 0.06, C21 = 0.05, c,2= 0.05, c„= 0.04, 

 Cm = 0.04, C2S = 0.04, C2« = 0.03, c^, = 0.03, C28= 0.03, 0,9= 0.03, c^= 0.03, 

 C3, = 0.02, C32 = 0.02, cas = 0.02, C34 = 0.02, 035= 0.01, C36== 0.01, and so 

 forth. These values are shown in figure 76. 



The way in which an inharmonic appears in the results of a 

 harmonic analysis is thus utterly different from that for a harmonic. The 

 third harmonic, p=3k, for example, appears in the analysis with different 

 values for a, and 63 (depending on the phase), but with for every other a 

 and b; it thus appears in its proper place with its full value and has no 

 influence anywhere else. The inharmonic, however, has no place of its 

 own in the series, but appears in every element. The average ordinate C 

 for a harmonic gives the height of the axis of the curve ; for an inharmonic 

 it depends not only on the height of the axis but also on the fractional 

 part of a wave included in the fundamental. 



When several inharmonic sinusoids coexist, the curve of their sum 

 will when analyzed yield a harmonic series of sinusoids whose ampUtudes 

 are the same as the ampUtudes of the original components. Thus for 

 the curve a', sin p't +a".sin p% we have 



a\ sin p't + a". sin p"t =C+ai.cos kt-\-ck-(ioa 2kt-{- . . . 



+ fti.sin kt + ftij-sin 2kt+ . . . 



where 



2£ air 



a'k r . .. . , a"k ' 



C = H /sin p't.dt + ^/sin pU.dt = C'+ C" , 



2t 3r 



a'k?. ' 



a, k C • n"k C 



^» = — J sin p'i. cos ikt.dt+ — J sin p"t. cos ikt.dt = a/'+ a", 



7t Tt 



air 2» 



a'k ~' 



0, k C ol'k C 

 hi = — J sin p't sin iktdt -\ J sin p"t. sin iktdt = 6/ + 6/'. 



(i=l,2, 3, . . .) 



From these equations the values of the coefficients can be calculated 

 as above. 



