,9o8.] KENNELLY AND UPSON— HUMMING TELEPHONE. 361 



receiver-diaphragm vibration ; so that for a fixed tube-length, 

 R==bm^r; or x(^')=^^V^; where b is the numerical constant 

 1.036 X 10" dynes per gm. and per Vcm. ; then for the example 

 already considered, if r^ioo; or y= 1,000, we find: 



The displacement r = o.oi cm. when ^ = 270°. 



The reinforced angular velocity is w = 5,184 radians per second; 

 « = 825 .— '. 



The maximum cyclic values of the vibratory — 



velocity v^ 51-84 cm. per sec. 



acceleration c = 268,740 cm. per sec.^ 



damping force Of= 5,184 dynes. 



restoring force OT = 5,184 dynes. 



elastic force OF' = 13,435 dynes. 



inertia force Of = 13,435 dynes. 



As the phase 6 oi reinforcement changes from 180° to 360°, the line 

 of y=^ 1,000 in Fig. 17 shows the change in frequency; while the 

 dotted line indicates the computed amplitude of vibration, which 

 reaches a maximum near 280°. 



According to the theory, therefore, if the phase of the displace- 

 ment is 270° behind the displacement of the receiver diaphragm, the 

 reinforced frequency coincides with the natural frequency. This 

 condition is substantially borne out in all of the observations. For 

 example, in Fig. 2, taking the pitch line No. i, with a natural fre- 

 quency of iIq = 825 -— ' and a sound-velocity in air of 33,000 cm. per 

 sec, the wave-length A ^ 33,000/825 = 40 cm. corresponding to 

 360° of phase. A lag of 270° would be represented by 30 cm. ; so 

 that we should expect the reinforced frequency to be 825 — ' at 30 

 cm. of tube-length, and at every 40 cm. beyond ; i. e., in accordance 

 with the series 30 -\- Apni, as was substantially observed. Moreover, 

 by reversing the receiver terminals, the phase of the reinforcement 

 is necessarily changed 180° ; so that with this change of connec- 

 tion, 270° of phase lag would be altered to 90° of phase lag, or 10 

 cm. of tube-length. The natural frequency of 825 '— ' should then 

 occur in conformity with the series 10 -\- 40m cm., as was substan- 

 tiallv observed. 



