THEORY 01' THE HOWLING TELEPHONE 37 



This is equi\alent to two scalar equations and taken together with 

 (19) and the curves of Fig. G gives the necessary four etjuations to 

 soK'c for the unknowns/. //, T, and /. 



The sohition, howe\er, is not straightforward since the ri-hi- 

 tion between //, 7", and / is only given empirically by a set of 

 (ur\es. By "cut and tr>" methods the solution for any numerical 

 case can be obtained. The last term of (20) is usually negligible 

 or at least it is of second order of magnitude. Consequently, the 

 sum of the phase angles of the other factors must be approximately 

 equal to the phase of Z. This completes the formal solution for 

 this case. 



The solution of a numerical case throws considerable light upon 

 the physical phenomenon taking place, and also upon the method of 

 calculation. Let the arm ratio be unity, a case corresponding to that 

 when the diaphragms are connected directly together, and assume 

 that the supply current is furnished by a battery of 24 volts through 

 a line having a resistance of 300 ohms. Using the constants for the 

 receivers and transmitters given above and expressing/ in kilocycles, 

 T in ohms, / in amperes and h in ohms per micron, equations (19) 

 and (20) become 



24 

 /=__^_^^ (19') 



384 + r' ^^ 



Ih o2|24°=[393 + r+r)0/+j(43 + 150/)] [-2.14/^+ 



2.3+J.14/1+/ 1.7/. (20') 



If / is positi\'e there is no solution for/, since the angle of the first 

 factor is in the first quadrant, and that of the second factor either 

 in the first or second; consequently, the phases cannot match at any 

 frequency. If the supply current is reversed, then / is negative or 

 180° is added to the phase of the left hand member making it a positive 

 150°. The solution for this case is 



/= 1072 cycles i = 8.2 mils 



A = 64 e = 5.5 volts 



r = 150 ohms y=z = 1.9 microns 



7 = 45 mils 



