THEORY OF VIBRATION OF THE LARYNX 225 



Suppose in (38) that A = \; then without loss in generahty: 



pz ^ Bp"' + Cp + D = 0. (47) 



In such an equation the roots are continuous functions of the co- 

 efficients. The same is true of their derivatives except at the one 

 point where transition occurs from pure real to complex values. The 

 values of the roots of interest in this connection are in their complex 

 region at the point where the real part of the root passes through a 

 zero value. This is the point at which free oscillation of the oscil- 

 lating mode occurs, the values of the roots of this mode being as shown 

 before, ± jwo. 



If it now be supposed that one cause produces small variations, 

 directly or indirectly on each of the coefficients and that the magni- 

 tude of this cause be .r, then : 



(3,= + 2., + C)g + .= f + .f + f = 0. (48) 



The problem then is to determine dp resulting from any assigned 

 cause dx when p = jooo- From (43) we have at this point B = Aq, 



C = coo" and D = Aqcoo^. 



.Ao\ dp ,dB , . dC , dD 



This is the frequency (complex) variation equation taken in the 

 neighborhood of free oscillation. 



When any readjustment of the larynx takes place all of the "con- 

 stants" entering the coefficients undergo change, in particular those 

 of the glottis Ku, Ri, G. Suppose for simplicity that one only varies, 

 then this variation dKu, dRi, or dG may be taken as the magnitude 

 of the cause dx. In particular if Ku vary, 



dB = = dC and dD/dx = G/L^L^, 

 .Ao\ dp G 



coo / dKu IwrfLiLo 



If in addition Ao be small compared with coo, 



. GdKu /, , .Ao\ 



dp = -> or J- 1 +J— • (.-^1) 



This shows that if a condition of sustained oscillation is departed 

 from by slightly increasing Ku, an increase in the amplitude of vibra- 

 tion begins which is proportional to the logarithm, since {p -\- dp) 



