394 Dr. Oliver Lodge on Opacity. 



or for a simple harmonic disturbance, 



g= (B,+yL0(Q 1 + vS,)V . . . (11) 

 = (*+WY, 



whose solution therefore is 



V = V e~ ax cos (pt-px) *...". (11') 



There are several interesting special cases : — 



The old cable theory of Lord Kelvin is obtained by omitting 

 both Q and L; thus getting equation (2). 



The transmission of Hertz waves along a perfectly-con- 

 ducting insulated wire is obtained by omitting Q and R ; the 

 speed of such transmission being 1/^(L 1 S 1 ). Resistance 

 in the wire brings it to the form (9), where the damping- 

 depends on the ratio of the capacity constant RS to the self- 

 induction constant L/S ; because the index R/2Li; equals half 

 the square root of this ratio ; but it must be remembered 

 that R has the throttled value due to merely superficial 

 penetration. The case is approximated to in telephony 

 sometimes. 



A remarkable case of undistorted (though attenuated) 

 transmission through a cable (discovered by Mr. Heaviside, 

 but not yet practically applied) is obtained by taking 



R/L = Q/S = r ; 

 the solution being then 



rx / T \ 



▼--•/HJ- 



due to fit) at #=0. All frequencies are thus treated alike, 

 and a true velocity of transmission makes its reappearance. 

 This is what he calls his ''distortionless circuit," which may 

 yet play an important part in practice. 



And lastly, the two cases which for brevity may be 

 treated together, the case of perfect insulation, Q = 0, on 

 the one hand, and the case of perfect wire conduction, R=0, 

 on the other. For either of these cases the general expres- 

 sion 



.W =te .L4{ I + (f L )*}'{ 1+ (J)'}'-{J?S-'}] 



* I don't know whether the following simple general expression for 

 a and /3 has been recorded by anyone : writing E/j9L=tan e and Q/^S = 

 tan e', 



' p sin or cos K*+0 nt'n\ 



a or £ = L - . — 2 ' J , (11 ) 



v (cose cos e)i 



which is shorter than (12). 



(12) 



