106 THE STUDY OF SPEECH CURVES. 



The coefficients are thus analogous to those given in equations (18, 

 19, 20) of Chapter V. The method of procedure is at once apparent. 

 The ordinates of ya,yi,y„ . ■ ■ y m-i are to be multiplied by 1, e'", e'"", 

 g(m-i)A.^ . . . respectively. The results are then treated just as the origi- 

 nal ordinates in the simple sinusoid analysis. 



For this, from a frictional sinusoid, equation (1) above, the value of 

 £ can be calculated if the successive amplitudes are known. It is readily 

 shown that the successive maximum amphtudes a^, ai, Ui, . . . in such 

 a curve bear the relations (+ and — disregarded) 



Oa a 



3 



T 



. e — 



2 



or 



T 

 log, oj — log, ai = log, oa - log, Oj =...=-£-=_ d. (H) 



The value d is called the logarithmic decrement; when it has been 

 obtained, the factor of friction e can be calculated. It is not necessary 

 to measure from the axis of the curve; the successive points are more 

 convenienth' measured from a line parallel to the axis; the results Si, 

 Si, S3, . . . yield Ukewise log, «2 = log, Si=log, 82= . . . = — d. 



The calculation according to the method of least squares gives the 

 familiar formula 



.^g j n-l) ilogeSi-logeS„)+(n-3) (logeSi-\oSeSn->)+{n-5) (logtSi-logeSn--^+ ■ . . (]9) 



n (»r-l; ^^ "'^ 



from which we obtain 



2d 

 T 



Examples will be given in the last chapter. 



