i5o8.] KENNELLY AND UPSON— HUMMING TELEPHONE. 353 



where ;'= \/ — i, and | is the displacement of the particle in cms. 

 from O at the angle <at, measured positively, or counter-clockwise, 

 from the initial line OX. 



Let the particle be acted upon by a centrally directed elastic force 



F ■= — A^^ — ma$^ — nian^'^^ dynes Z (2) 



proportional to and opposing the displacement, as represented by 

 the vector OF in Fig. 14. Let there be no other forces except 

 those of inertia, acting on the particle ; so that the movement is 

 frictionless. Then the velocity of the particle at any instant t will be 



v = $^ j(ore^"^ cms./sec. Z (3) 



The direction of the velocity will, therefore, be perpendicular to the 

 radius vector, or parallel to the instantaneous tangent, as indicated 

 by the dotted line Ov, 90° ahead of Os in phase displacement. 

 ■ The acceleration of the particle will be, at any instant t, 



c = v = i^ — w-re^'w* cms./sec- Z (4) 



That is, the acceleration will be directed oppositely to the displace- 

 ment. Thus at time ^ = 0, represented in Fig. 14, the acceleration 

 w'ill be directed along OY. The virtual reactive force of inertia 

 will be 



/ = — mc = — mi = niw-n^"'^ dynes Z (5) 



In Fig. 14, this reactive force of inertia is represented by Of. 



In order that the circular orbital motion shall be stable, the 

 sum of the forces OF and Of, of elasticity and inertia must be zero; 

 or 



OF -^Of = o dynes Z 



whence 



— mare^'^^ -\- niurrt^'^^ = o dynes Z 



= \/A/m = V^ radians/sec. (6) 



