Potentiometer on Alternate Current Circuits. 407 



that it is the fundamental wave only which is measured, and 

 the comparison is really between the mean and not the 

 R.M.S. values. On the other hand, if there is any serious 

 difference of wave-form, the sensitiveness to the upper 

 harmonics should be sufficient to prevent an exact balance 

 being- secured, and this warns us when great accuracy is not 

 to be expected. 



The effective value of an irregular periodic voltage is 



V = \/% TV, 



where V tl is the effective value of the harmonic of order n. 

 If the sum of the squares of the higher harmonics is small 

 in comparison with the square of the fundamental, we may 

 take the approximate square root, and 



1 



v = v 1 + 2Vi ^v 



or 



v = Vl { 1 + -nV- vV }' 



where 2V„ 2 in the last two expressions implies the sum of 

 the squares of the effective values of all the harmonics, 



excluding the fundamental. 



In the case of a single harmonic this becomes 



*M^m- 



and the expression in brackets oily differs 1 per cent, from 

 unity when Y n is II per cent, of Vj. Such a difference 

 should be fairly easily recognizable, and would probably be 

 sufficient to prevent the obtaining of a perfect balance on the 

 vibration galvanometer ; and, moreover, if the distortion of 

 wave-shape is great, there is little object, as a rule, in great 

 accuracy of measurement. If accuracy is required in such 

 cases, it can of course be secured by tracing the wave-shape 

 and making the above correction, but if care is taken to have 

 a supply of fairly sinusoidal wave-form, the measurements 

 may be considered sufficiently accurate for the majority of 

 purposes. It should be observed that this difficulty, or a 

 similar one, is inseparable from any method which can be 

 devised, in which the detector is employed to indicate the 

 difference between the test and potentiometer P.Ds. as is 

 done with direct currents. 



Measurements. — Two illustrations will be sufficient to show 



