94 THE STUDY OF SPEECH CURVES. 



When the period of the curve analyzed is found in the harmonic series, 

 ; is a whole number; the numerators of the fractions are in every case 

 = ; the denominators are concrete numbers for all cases except j = i; 

 consequently the coefficients C, a,, Oj, . . . 61, b^, . . . except a,, 6„ are 

 also = 0. For a, and 6, the denominators are = 0, and the fractions are inde- 

 terminate. We have, however, 



"> = -•/ sin jkt. cos jkt.dijkt) = z^- [^'"' ?^^] ^^' 

 and 



2r 

 h 



^ = ^-j sin' jkt.d{jkt)=^a. 



Thus the analyzed curve appears in the harmonic series with its full 

 amplitude for the member having the same period and phase, and with 

 zero for all other members. When the period of the curve analyzed is not 

 to be found in the harmonic series, ; is not a whole number. Here the 

 vibration has a frequency that is not an even multiple of the lowest term 

 of the series, that is, its period does not occur in the harmonic series. Such 

 a curve is " inharmonic" to the series used for the analysis. The harmonic 

 coefficients are given by the formulas (27, 28). For example, the curve 

 o.sin pi, where p = 3^A, will appear in the analysis with the coefficients, 



C ^^^^ = hlUl ak\ 



di 



_ 7a.sin' Sl^r _ a 



~ Ti\ (3if — r' I ~ 5.4978-0.4488 i' ' 



7 _ oi&lsin 7,T _ q 



'~ iz\ci\f~'e\~' 



Expressed as a harmonic series of sines the curve has the coefficients 

 = " f. a _ « _ a _ a 



5.0490' "''"3.7026' ' 1.4576' "-^ -1.6830' '""-5.7222' 

 and so forth, with the factors of phase 



Qi = q2 = q3 = ^,qi = q,^ ... =~. 



