MECHANICS OF THE INNER EAR 85 



Let US combine two sinusoids according to the following 

 equation : 



/(;*:) ^ 2sin;i:-f-sin2;r. 

 The combination ^^^* ^^' ^^^ amplitude of the sinusoid of 

 1 and 2, when 2 the shorter period is one-half of the am- 

 is comparatively plitude of the sinusoid of the longer pe- 

 weak riod. Figure 26 shows the curve represent- 



ing the stirrup movement, and the accom- 

 panying table shows the exact numerical values of those points 

 of the curve which, as we shall see, are of particular import- 

 ance to us, that is, the maxima and minima, and the points 

 of inflection. These values are easily found in this particular 

 case. To find the maxima and minima, we have to set the 

 first derivative of the above function equal to zero and solve 

 the equation for x; for the maxima and minima are those 

 points where the tangential angle or differential coefficient 

 is zero. 



f'(x) = 2cos;tr -f- 2cos2a: = 0. 



To find the points of inflection, we have to set the second 

 derivative equal to zero and solve the equation for x; for the 

 points of inflection are those points of the curve where the 

 tangential angle neither increases nor decreases. 

 f"(x) = — 2sin.tr — 4sin2x = 0. 



The purely arithmetical work I do not care to perform 

 here. The table shows its results. It is plain that, if we rep- 

 resent the successive positions of the partition according to 

 the same rules as formerly employed, we find that only one 

 tone can become audible, the tone 1. The tone 2 has disap- 

 peared because its addition does not increase the number of 

 the maxima and minima of the compound curve (Fig. 26), but 

 merely influences its shape. However interesting this in- 

 sight may be into the fact that a weak higher tone added 

 to a strong lower tone may be entirely inaudible, the present 

 theoretic result is not quite satisfactory. It is somewhat un- 



