310 ME. W. DUDDELL ON THE RESISTANCE AND 



which the resistance is a function of the conditions, the possibility of obtaining the 

 power-factor unity is a proof of the constancy of the resistance and consequently of 

 the conditions, so that if the apparatus A is an arc, and if it can be shown that a 

 sufficiently high value of the frequency can be reached for which the power-factor is 

 unity, then the conditions of the arc are not being altered by the alternating testing 

 current, and the arc has a true resistance numerically equal to I A . 



It is assumed above that the arc or apparatus A has no self-induction or capacity ; 

 to prove this it must be shown that not only can the frequency be increased till the 

 power-factor becomes unity, but also that it remains so for a considerable further 

 increase of frequency.* 



Finally, therefore, in order to prove that the arc has a true resistance, and to find 

 its value, it is necessary to show : First, that it is possible to find a value of the 

 frequency of the alternating testing current for which the power-factor of the arc 

 with respect to this current is unity ; second, that the power-factor remains unity 

 and the impedance constant even when the freqiiency is greatly increased above this 

 value ; third, to determine the value of the impedance of the arc under these 

 conditions, which will also be its true resistance. 



Method of Measuring the Impedance and Power-Factor. 



At 'first sight it would seem as if there were a considerable number of available 

 methods for accurately measuring these quantities. But the number of methods 

 becomes exceedingly limited when it is considered that it is necessary for the 

 alternating testing current C to have as small a KM.S. value as possible (O'l ampere 

 was that generally used in the experiments), and that the effects due to this small 

 current have to be sorted out when it is mixed with a direct current of 10 amperes 

 or more. Added to this, to make the difficulties greater, it was finally found 

 necessary to use frequencies up to and even over 100,000 per second. Watt- 

 meters and dynamometers were tried and abandoned, and finally the well-known 

 3-voltmeter methodf was adopted. 



A non-inductive resistance R was placed in series with the apparatus A (fig. 2), 

 through both of which the main direct current flowed ; to this direct current there 

 was added, as before, an alternating measuring current of R M.S. value C. 



Let V A , V B , and V be the E.M.S. values of the alternating part of the P.D.'s as 

 shown in fig. 2. The impedance of A is I A = V A /C = RV A /V E . Power factor of 

 A is P A = (V 2 - V A 2 - V B 2 )/(2V A V E ). 



It seems possible that the power-factor of a conductor which did not possess self-induction or resistance 

 in the ordinary sense of these terms might still depart from unity at very high frequencies, owing to the 

 time taken by the carriers of the electric charge to hand it on becoming comparable with the periodic 

 time of the testing current. 



t See AYRTON and SUMFNER, ' Roy. Soc. Proc.,' vol. 49, p. 424. 



