354 KENNELLY AND UPSON— HUMMING TELEPHONE. [Ju.y 20, 



If, therefore, the angular velocity of the motion be numerically 

 equal to the square root of elastic force per unit of mass, the 

 orbit will be circular and stable, and Fig. 14 may represent its 

 vector diagram. The particle z rotates about 0, at constant radius 

 with uniform angular velocity oj, and the pair of equilibrating 

 forces OF and Of rotate in synchronism with it. The entire 

 system, Fig 14, may be imagined as pivoted about an axis through 

 O perpendicular to the orbital plane, and spun about this pivot with 

 uniform angular velocity w. 



By a well known proposition connecting simple harmonic vibra- 

 tion with circular orbital motion, the displacements in the former 

 are the projections of the displacements in the latter, upon a straight 

 line passing through the center of the system. In other words, to 

 every case of simple circular orbital motion in two dimensions 

 corresponds a case of simple harmonic vibration, its projection in 

 a single dimension. Consequently, at time t, we have for the dis- 

 placement in the case of simple vibration, 



I = re^wt cms. {y) 



measured along the initial line OX by projection. The real part 

 only of I is retained, and the imaginary part ignored. Similarly, 

 the vibratory velocity will be 



v = ^ = juyre^^* cms./sec. (8) 



taking only the real part of the equation, or the projected value 

 along YOX. Again, the vibratory acceleration will be 



c^ — w-i'e''^^ cms./sec.^ (9) 



retaining only the real or projected part. Similar reasoning ap- 

 plies to the forces of elasticity' and inertia. The same equations 

 appear as in the circular orbit case ; but only their real, or horizon- 

 tally projected values, are retained. Consequently, we deduce that 

 the vibration of a particle possessing elasticity and inertia without 

 frictional retardation will be stable and self sustained under the 

 condition 



