i9o8.] KENNELLY AND UPSON— HUMMING TELEPHONE. 359 



or 



— iiiare'^^ -\- nix(r) cosd e^'^^ -{-niw-re^'^^^o dynes Z 



whence 



CO = Va — cos(9 • X (^) A radians/sec. (17) 



Considering the velocity pair, the first member is 



Of = — Tv = — 2myv = — J2inywre^^* dynes Z 



The second member is the T component : 



OT^ — jnixir) sin^ • e^''"'* dynes Z 



For equiHbrium 



Of' + OT = o 

 or 



— J2rny(iirei<^^ — jmx (r) s'mO ■ t^^^ = o 

 From which 



x{r) sin 6 = — 2ryoj dynes (18) 



and 



o) = V«o^ + 7" cot- 6 -\- y cot d radians/sec. (19) 



It follows that w. the new angular velocity under reinforcement, 

 is independent of the force function R = nix{r), and depends only 

 on the natural angular velocity wq, the phase retardation 6 of the re- 

 storing force and the magnitude of the damping coefficient y. Some 

 curves of w as a function of 6 for four particular values of V be- 

 tween 50 and 500 dynes per cm. per sec. ; i. e., of y between 500 and 

 5,000 dynes per cm. per sec. and per gm., are given in Fig. 17. It 

 may be seen that for all values of the damping, a)==wo for ^^270°. 

 That is, the angular velocity of reinforced motion is the same as 

 that of unretarded motion when the restoring force is applied at 270° 

 of phase lag, or exactly in phase with the velocity, as seen in Fig. 

 16. If the phase retardation ^ is between 180° and 270°, the new 

 angular velocity will be greater than the natural angular velocity wq ; 

 but if B is between 270° and 360°, must be less than wq. 



Applying the above principle to the corresponding case of rein- 

 forced vibration, by taking the projections or real parts of the rotat- 



