Tow Frequency and their Resonance. 427 



confined to the condenser, for the voltage on the line, indica- 

 ted by the Cardew Voltmeter, does not change sensibly when 

 resonance is established. There is a large and rapid change 

 in the current with the approach of resonance which can be 

 studied in a rough way by the pull which the choking coil 

 exerts upon the removable iron core when the core is moved 

 up and down during the process of tuning. The variation of 

 this pull indicates very plainly that the curve expressing the 

 relation between the current and the self-induction (resistance, 

 capacity and frequency being constant), has a very steep crest 

 which is in perfect accordance with the carefully plotted curve 

 of equation (7) in Bedell and Crehore's volume on alternating 

 currents.* 



There are, however, several large maxima in this curve, each 

 corresponding to a different capacity and self-induction • the 

 simple experiment just described shows their presence very 

 forcibly. The maximum corresponding to the largest capacity 

 with about the same self-induction being however consider- 

 ably the highest. With the condensers that I had at my dis- 

 posal at that time I did not dare to tune the circuit for the 

 highest maximum. The existence of several maxima will be 

 seen presently to be a necessary consequence of the theory. 



IV. Electrical Resonance in a Circuit with a Complex 

 Harmonic Electromotive Force. 



By Fourier's theorem a complex harmonic alternating e. m. 

 f, can always be represented by the following series : 



Err a x sin pt + a 2 sin 2pt + .... a„ sin npt .... 



cc 



= 2 a aa sin orpt 

 i 



In this expression I shall call a 1 sin pt, a 2 sin 2 pt, .... the 

 component harmonics, a 1 sin pt is the \ fundamental harmonic, 

 its frequency, the fundamental frequency. The other harmon- 

 ics will be referred to as the upper harmonics. The order of 

 magnitude of their amplitudes is a x > a z > a 3 > . . . . > a n > . . . . 



The symbolical expression of Ohm's law is this : 



x dx ._ _ « 



L — \- ixx + r= 2;a aa Sin apt 



at i 



Comparing this to (6) it is seen from the integral in (7) that 

 this differential equation has the following expression for its 

 integral : 



*See Bedell and Crehore's treatise: Alternating Currents, p. 138, published by 

 W. J. Johnston Co., New York. 



Am. Jour. Sci.— Third Series, Vol. XLV, No. 269.— May, 1893. 

 30 



