Forced Vibrations of Electromagnetic Systems. 209 



function at the sending end. Suppose the apparatus to be 

 representable as a resistance containing an electromotive 

 force, and that by varying the resistance we cause the electro- 

 motive force to vary as its square root. Then, according to 

 a well-known law, the arrangement producing the maximum 

 external current is given by R = Ly, equality of impedances 

 again. This brings us to 



as if the circuit were infinitely long both ways, with maximum 

 efficiency secured at both ends. 



Lastly, the choice of L such that RZ = 2Lu makes the circuit, 

 of given resistance, most efficient. 



In long-distance telephony using wires of low resistance, 

 the waves are sent along the circuit in a manner closely re- 

 sembling the transmission of waves along a stretched elastic 

 cord, subject to a small amount of friction. In order to 

 similarly imitate the electrostatic theory, we must so reduce 

 the mass of the cord, or else so exaggerate the friction, that 

 there cannot be free vibrations. We may suppose that the 

 displacement of the cord represents the potential-difference in 

 both cases. But the current will be in the same phase as the 

 potential-difference in one case, and proportional to its variation 

 along the circuit in the other. 



We may conveniently divide circuits, so far as their signal- 

 ling peculiarities are concerned, into five classes. (1) Cir- 

 cuits of so short length, or so operated upon, that any effects 

 due to electric displacement are insensible. The theory is 

 then entirely electromagnetic, at least so far as numerical re- 

 sults are concerned. (2) Circuits of such great length that 

 they can only be worked so slowly as to render electro- 

 magnetic inertia numerically insignificant in its effects. 

 Also some telephonic circuits in which R/Ln is large. Then, 

 at least so far as the reception of signals is concerned, we 

 may apply the electrostatic theory. (3) The exceedingly 

 large intermediate class in which both the electrostatic and 

 electromagnetic sides have to be considered, not separately, 

 but conjointly. (4) The simplified form of the last to which 

 we are led when the signals are very rapid and the wires of 

 low resistance. (5) The non-distortional circuit, in which, 

 by a proper amount of uniform leakage, distortion of signals 

 is abolished, whether fast or slow. Regarded from the point 

 of view of practical application, this class lies on one side. 

 But from the theoretical point of view, the non-distortional 

 circuit lies in the very focus of the general theory, reducing 



