THE ELECTRIC WAVE-FILTER 11 



above, and the surging is localized within the section; twice during 

 each cycle the amount of energy increases on the right and decreases 

 on the left. But we do not know that the section is made up like 

 Fig. G; we only know that it is ecjuivalent to Fig. 6 as regards input 

 and output relations. As far as these external relations go, the actual 

 network may be made exclusively of either inductances or capacities 

 with the connections shown in Fig. 4 or with the cross-connections 

 of Fig. 5, according as the current is to have the same or opposite 

 signs in consecutive sections. In any network made up exclusively 

 of inductances or of capacities, the total energy falls to zero when 

 the current or the potential falls to zero, respectively. Twice, there- 

 fore, in every cycle the total energy surges into this line and then it 

 all returns to the generator. With other networks, surgings inter- 

 mediate betw'een these two extremes will occur. The theorem, 

 therefore, does not limit the extent of the surging. 



Under the second assumption, the phase difference between the 

 currents at two given points, separated by a periodic interval, is to 

 be an angle which is neither zero nor a multiple of ±x. The assumed 

 difference in phase can only be due to the infinite extension of the 

 artificial line since, as previously noted, no finite sequence of induct- 

 ances and capacities can produce any difference in phase. That 

 infinite lines do produce phase differences is well-known; in particular, 

 an infinite, uniform, perfectly conducting, metallic pair shows a 

 continuous retardation in phase. If the infinitely remote sections 

 of the artificial line are to have this controlling effect oti the wave 

 motion, the wave motion must actually extend to infinity, that is, 

 there can be no attenuation. The wave progressing indefinitely to 

 infinity without attenuation must be supplied continuously with 

 energ\'; this energy must flow along the entire line with neither loss 

 nor gain in the reactances it encounters on the way. This continuous 

 flow of energ>' can take place only provided the currents and poten- 

 tials are not in quadrature; they may be in phase. In considering 

 the free oscillations of Fig. 6 it was shown that K2 is real if it is not 

 pure reactance. That is, for the mid-shunt section the current and 

 potential are in phase. It is easy to show that they are also in phase 

 at the mid-series point which is also a point of symmetry. 



This flow-of-energy state of motion thus necessarily characterizes 

 a phase-retarded wave on a resistanceless artificial line, regardless 

 of the amount of the assumed positive or negative retardation, which 

 may be taken to have any value betw'een zero and exact opposition 

 of phase. Continuously varying the retardation throughout the 180 

 degrees will, in general, call for a continuous change in the frequency 



