THE ELECTRIC IFAVE-PILTER 9 



let us consiticr tlic free oscillations of I^'ig. 0; first, with A'o assumed 

 to be a pure reactance; second, with A'2 assumed to be a pure resist- 

 ance; and third, in order to show that this third assumption is con- 

 trary to fact, with A'o assumed to be an impedance with both resist- 

 ance and reactance. 



With Ao a reactance, the circuit contains nothing but reactances, 

 and free oscillations are possible if, and only if, the total impedance 

 of the circuit is zero. The end impedances Z' and Z" being different, 

 the potentials at the ends of the mesh will be dilTerent, and this means 

 that the corresponding wave on the infinite line will be attenuated, 

 since the ratio between these potentials is the rate at which the am- 

 plitudes fall ofY per section. 



With Ao a pure resistance, a free oscillation is possible only if the 

 dissipation in the positive resistance at the right end of the circuit 

 is exactly made up by the hypothetical source of energ\' existing in 

 the negative resistance — A2 at the left end of the circuit. An exact 

 balance between the energy supplied at one end and that lost at the 

 other end is possible, since the equal positive and negative resistances 

 A2, — Ao carry equal currents. This continuous transfer of energy 

 from the left of the oscillating circuit of Fig. 6 to the right end is the 

 action which goes on in every section of the infinite artificial line, and 

 serves to pass forward the energy along the infinite line. 



If Ao were complex, — Ao on the left of Fig. 6 and +A2 on the 

 right would not carry the same fraction of the circulating current /, 

 since they are each shunted by a reactance 2Z2 which would allow 

 less of the current to flow through +A2 than through — Ao, if 2Z2 

 makes the smaller angle with +A2, and vice versa. No balance 

 between absorbed and dissipated energy is possible under these con- 

 ditions when the equal and opposite resistance components carry 

 unequal currents. A complex A2, therefore, gives no free oscilla- 

 tion, and cannot occur with a resistanceless artificial line. 



It is perhaps more instructive to consider the transmission on the 

 line as a whole, rather than to confine attention exclusively to the 

 oscillations of the simple circuit of Fig. 6 and so, at this point, w^ith- 

 out following further the conclusions to be drawn directly from this 

 oscillating circuit, the fundamental energy theorem of resistanceless 

 artificial lines will be stated, and then proved as a property of an 

 infinite artificial line. 



Energy Flow Theorem 



Upon an infinite line of periodic recurrent structure, and devoid of 

 resistance, a sinusoidal e.m.f. produces one of two steady states, viz.: 



