356 KENNELLY AND UPSON— HUMMING TELEPHONE. [July 20, 



Each of the above equations defines an equiangular spiral, an in- 

 wardly directed spiral in which the curve makes a constant direction 

 — y -\- jo) with the radius vector. 



The vector diagram for this case is indicated in Fig. 15. Let 



Y' 



Fig. 15. Vector Diagram of Free Damped Vibration. 



2 be the position of the particle at any instant. The velocity at this 

 instant will have the vector OV, parallel to the tangent at z, where 



tan cj> = (o/y (15) 



The acceleration at the same instant will be directed along OY, 

 the angles XOV and VOY being each equal to the supplement of ^. 

 The virtual force of inertia will be directed along Of. The retard- 

 ing force, opposing the velocity, will be directed along Of. At any 

 instant the vector sum of the three forces of elasticity, retardation 

 and inertia must be zero. That is. 



or 



OF -f O/' -f 0/ = o dynes Z 



— mare'-y^^^^* — 2ynir{ — y -f yo))e^-7+-'"w)« 

 — uirQ— y -\- yc.j)-e'-7+^''<')< :r= o dynes Z 



whence 



= V^ — y^=Vwo" — y' = ^o sin (^ radians/sec. (16) 



where Wg is the unretarded angular velocity. That is, the angular 

 velocity of orbital rotation has been reduced by the retardation in 

 the ratio of sin ^, Fig. 15, and the displacement or radius vector r 

 continually dwindles with time by £~7'. 



In the corresponding case of free damped vibration, the above 



