448 KENNELLY, TAYLOR— PROPERTIES OF 



Finally from (13), we obtain 



dyne perp. cm. 



5 = 1.006 X 10"'' X 6286- = 397.5 T- • 



radian 



The reason for making the diameter Or of the deflection graph 

 lag 90° behind the diameter OR of the impedance circle, and the 

 diameter OR' of the velocity circle, is that according to (4), the 

 phase of any displacement is 90° behind the corresponding velocity 

 0. Strictly speaking, while the velocity graph is a circle ; the de- 

 flection graph is only approximately a circle, since in (4), the vari- 

 able oj appears directly as a factor in the denominator. The angular 

 velocity of maximum deflection to,/, is not the same as the resonant 

 angular velocity w^, but is 



; — 7, n radians ^ . 



Wrf = Vcoo- - 2A^ "^e^" ' ^^"^^ 



while the angular velocity of free oscillations wf, in the presence of 

 damping, lies between ioa and w^ ; namely 



I — „ , radians , , 



cof = Vcoo-- A^ . 18) 



sec. 



In this case, taking lo^ = 6,286.0, w/= = 6,286.06 and m = 6,286. 1 3 ; 

 or the damping is so small, and the resonance is so sharp, that the 

 difference between these three important and characteristic angular 

 velocities is very small. 



We may define the sharpness of resonance, or sharpness of tuning 

 (inverted V),^^ in relation to the resonant and quadrantal angular 

 velocities w^, oj^ and o)^ by the formula^ ^ 



COo COo Wo . , . 



A = = —7 = , numeric, (19} 



0)2 ~ toi 2A no — Wi 



which in this case is 6,286/56.6=111.1, a relatively high figure of 

 merit in regard to tuning.^** The reciprocal of A may perhaps be 



13 Suggested by Dr. R. L. Jones. Bibliography No. 8. The expression 

 " sharpness of tuning " has also been suggested by Barton (bibliography, 4) 

 with a different quantitative meaning. 



1* In the case of a certain experimental monopolar telephone receiver, 

 tested by Kennelly and Pierce (bibliography, 7), the sharpness of resonance 

 was found to be 161. 2, the highest experimental value in electromagnetic appa- 

 ratus yet observed in these researches. 



