10 BELL SYSTEM TECHNICAL JOURNAL 



1. A to-and-jro surging of energy without any resultant transfer 

 of energy; currents and potential differences each attenuated from 

 section to section, but everywhere in the same or opposite phase and 

 mutually in quadrature, or, 



2. A continuous, non- attenuated flow of energy along the line 

 to infinity with no energy surging between symmetrical sections; 

 current and potential non-attenuated, but retarded or advanced in 

 phase from section to section, and mutually in phase at mid-shunt 

 and mid-series points. 



The critical frequencies separating the two states of motion are the 

 totality of the resonant frequencies of the series impedance, the anti- 

 resonant frequencies of the shunt impedance, and the resonant frequencies 

 of a single mid-shunt section of the line. 



To prove the several statements of this theorem let us consider 

 first the consequences of assuming that the wave motion, in progress- 

 ing along the line, is attenuated, and next the consequences of assum- 

 ing that the wave motion changes its phase. If the wave is atten- 

 uated, however little, at a sufficient distance it becomes negligible, 

 and the more remote portions of the line may be completely removed 

 without appreciable effect upon the disturbance in the nearer portion 

 of the line. That part of the line which then remains is a finite net- 

 work of pure reactances, and in any such network all currents are 

 always in the same, or opposite, phase; so, also, are the potential 

 differences; moreover, the two are mutually in quadrature; there is 

 no continuous accumulation of energy anywhere, but only an ex- 

 change of energy back and forth between the inductances, the ca- 

 pacities and the generator. Continuously varying the amount of 

 the assumed attenuation will cause a continuous variation in the 

 corresponding frequency. The motion of the assumed character 

 may, therefore, be expected to occur throughout continuous ranges 

 or bands of frequencies and not merely at isolated frequencies. 



The question may be asked — How far does the energy surge? Is 

 the surge localized in the individual section, or does the surge carry 

 the energy back and forth over more than one section, or even in and 

 out of the line as a whole? To answer this question, it would be 

 necessary, as we will now proceed to prove, to know something about 

 the actual construction of the individual section. If each section is 

 actually made up as shown in Fig. 6, and this is entirely possible in 

 the present case (since only positive and negative reactances would 

 be called for) , then the section is capable of free oscillation, as explained 



