138 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the circul.ar graphs. The formulas by which this distribution has been 

 computed are derived under heading V below. 



The quantities of theoretical and practical importance in these 

 circular graphs are: 



1. The length of diameter of the circular graph for a particular 

 receiver. 



2. The dip of the diameter below the axis of resistance. 



3. The rate of change of angular velocities around the circle. 



4. The angular velocity at the end point of the diameter, remote 

 from the origin, and 



5. The impedance at this point. 



The significance of these several quantities will appear in connection 

 with the discussion of the theory of the problem, which follows. 



V. Theory of the Reactive Effects of Motion of the Dia- 

 phragm ON the Electrical Constants of the Receiver. 



An exact treatment of the electrical properties of a coil containing a 

 magnetic core in proximity to a moving magnetic membrane offers 

 great difficulty. If, however, we confine our attention to terms of the 

 first order, we can obtain a sufficiently close approximation to a solu- 

 tion, to permit an interpretation of the preceding experimental results. 



Assumptions Regarding Mechanical Magnitudes. — To this 

 end we shall assume, so far as concerns the fundamental mode of 

 vibration of the diaphragm, 



(1) That the elastic restoring force of the diaphragm is all concen- 

 trated at the center of the diaphragm, and is proportional to the dis- 

 placement; 



(2) That the motion is opposed by a frictional force proportional to 

 the velocity and also concentrated at the center of the diaphragm ; and 



(3) That the actual distributed mass of the diaphragm may be 

 replaced by an equivalent mass concentrated at the center of the 

 diaphragm. 



Motion of the Center of the Diaphragm under a Pull Main- 

 tained Sinusoidal. — As a first step toward the solution, let us 

 assume the diaphragm to be solicited by a force which is maintained 

 sinusoidal; then (cf. Figure 13) 



s.r + ra; + m'x = f = /V"' ^ dynes Z (5) 



2 The sign / following the unit indicates that the equation should be inter- 

 preted vectorially, or in complex quantities. 



