82 Mr. 0. Heaviside on the 



peculiarities connected with the looped-wire problem in ad- 

 dition to those described in Part IV., and to make the neces- 

 sary limitations of application of the method and the results 

 which are required by the presence of the earth. 



First of all, even though the wires be not connected to 

 earth, if they be charged and currented in the most arbitrary 

 manner possible, we must employ the four electrical variables 

 and the ten or eleven electrical constants as above mentioned. 

 On the other hand, going back to the looped wires far removed 

 from other conductors, there are but two electrical variables 

 and four constants (counting one for leakage). Now bring 

 these parallel wires to a distance above the earth which is a 

 large multiple of their distance apart. The constant S of 

 capacity is a little increased. The method of images gives 



S = c(21og!l!4)" 1 ; .... (3/) 



where r ly r 2 are the radii of the wires, r 12 their distance apart, 

 s 1} s 2 their distances from their images, and s 12 the distance 

 from either to the image of the other ; but, owing to s^/s^ 

 being nearly unity, the capacity S does not sensibly differ 

 from the value in an infinite dielectric, or the earth has scarcely 

 anything to do with the matter*. If, however, the wires be 

 brought close to the earth, the increase of capacity will become 

 considerable ; this is also the case when the wires are buried. 

 The extreme is reached when each wire is surrounded by 

 dielectric to a certain distance, and the space between and 

 surrounding the two dielectrics is wholly filled up with 

 well-conducting matter. Then the capacity S becomes the 

 reciprocal of the sum of the capacities of the two wires with 

 respect to the enveloping conductive matter ; in another form, 

 the effective elastance is the sum of the elastances of the two 

 dielectrics. Returning to the suspended wires, if the earth 

 were infinitely conducting, the effective inductance would be 

 the reciprocal of S in (3/) with jul written for c, in electromag- 

 netic units, with J (jii + f^) added ; whilst, allowing for the 

 full extension of the magnetic field into the earth, we should 



* On the other hand, Mr. W. H. Preece, F.R.S., assures us that the 

 capacity is halved (Proc. Roy. Soc. March 8, 1887, and Journal S. T. E. 

 and E., June 27 and Febr. 10. 1887). This is simple, but inaccurate. 

 It is, however, a mere trifle in comparison with Mr. Preece's other errors ; 

 he does not fairly appreciate the theory of the transmission of signals, 

 even keeping to the quite special case of a long and slowly worked sub- 

 marine cable, whose theory, or what he imagines it to be, he applies, in 

 the most confident manner possible, universally ! There is hardly any 

 resemblance between the manner of transmission of currents of great fre- 

 quency and slow signals. 



