Forced Vibrations of Electromagnetic Systems. 207 



It will be seen that when the resistance of the circuit is 

 only a small multiple of, or is of about the same magnitude 

 as hv (which may be from 300 to 600 ohms in the case of a 

 suspended copper wire), the variation in the value of p as 

 the frequency changes through a sufficiently wide range, is 

 great, merely by reason of the reflexions causing reinforce- 

 ment or reduction of the strength of the received current. 

 The theoretical least value of p is ^, when R/Ln is vanishingly 

 small, indicating a doubling of the amplitude of current. 

 But as y increases the range of p gets smaller and smaller. 

 After y = 5 it is negligible. 



It is, however, the mean p that is of most importance, 

 because the influence of terminal resistances is to lower the 

 range in p, and to a variable extent. The value y — 2*065, or, 

 practically, Ri = 2Lv, makes the mean p a minimum. As I 

 pointed out in the paper before referred to, these fluctuations 

 can only be prejudicial to telephony. In the present Note I 

 have described how to almost entirely destroy them. The prin- 

 ciple may be understood thus. Let the circuit be infinitely 

 long first. Then its impedance to an intermediate impressed 

 force alternating with sufficient frequency to make R/Ln small 

 will be 2Lv, viz. Lv each way. The current and potential- 

 difference produced will be in the same phase, and in moving 

 away from the source of energy they will be similarly at- 

 tenuated according to the time-factor e- R '/ 2L . In order that 

 the circuit, when of finite length, shall still behave as if of 

 infinite length, the constancy of the impedance suggests to 

 us that we should make the terminal apparatus a mere re- 

 sistance, of amount Lu, by which the waves will be absorbed 

 without reflexion. 



That this is correct we may prove by my formula for the 

 amplitude of received current when there is terminal appa- 

 ratus, equation (19 b), Part Y. " On the Self-induction of 

 Wires " (Phil. Mag. Jan. 1887). It is 



C ° = 2Y o[]FtS3 [G ° ai e2P '+ H o H i ''**' 



-2(G G 1 H H 1 )i cos 2(QZ + 0)] -*. 



Here C is the amplitude of received current at z = I due to 

 V sinn£ impressed force at z = 0; R/ and 1/ the effective re- 

 sistance and inductance per unit length of circuit ; K and S 

 the leakage-conductance and permittance per unit-length, 



P or Q=(A)M(R' 2 + LV)*(K^SV) i ± (KR'-I/Sn 2 )}*; 

 G , H , are terminal functions depending upon the apparatus 



