(28) 



132 Mr. 0. Heaviside on the 



through the dielectric, so that unless p be extravagantly large 

 fji 2 cp 2 is exceedingly small also. Thus, with moderate distance 

 of return-current, s 2 a 2 is in general exceedingly small. 



Therefore in the expressions (17) take first terms only, 

 making 



J (s 2 r) = l, Ji( s 2^)=iv; \ § (26) 



These, used in (25), bring it down to 



<:- j ^=4^ j °^) ; • • • < 27 > 



concerning which, so far as numerical accuracy is concerned, 



the only assumption made is that the return has no resistance. 



We. have now the following complete normal system : — 



Hi = A Ji(sir) cos (mz 4- 0)e^, 

 4:7ry 1 -= AJ 1 (s 1 ^)m sin (mz + 6)e pt } 

 ^wTi = AJ (s 1 r)s 1 cos (mz + 6)eP t , 



H 2 = B(^)-i cos (mz + $)&', 

 krry 2 — B(slr)-hn sin (mz + 6)eP*, 

 47rr 2 = B log (a 2 /r) cos (mz -f 0)eP t , 



where B = A(p 1 s 1 /p 2 ) J (s 1 a 1 )-r- log (« 2 /%)- 



The longitudinal current and electric force in the dielectric 

 vary as the logarithm of the ratio a 2 /r, vanishing at r = a 2 . 

 The radial components vary inversely as the distance. Nume- 

 rically considered, the longitudinal electric force is negligible 

 against the radial, which is important as causing the elec- 

 trostatic retardation on long lines. But, theoretically, the 

 longitudinal component of the electric force is very important 

 when we look to the physical actions that take place, as it 

 determines the passage of energy from the dielectric, its seat 

 of transmission along the wire, into the conductor, where it 

 is dissipated. 



Regarding (28), however, it is to be remarked that, on 

 account of the approximations, the dielectric solutions do not 

 satisfy the fundamental equation (6). Applying it, we get 

 r = 0. But the other fundamental (7) is satisfied. To satisfy 

 (6), take 



KiM = — M ~ l + is 2 r (log s 2 r— 1) ; 

 leading to the determinantal equation 



iogj. j^f **>*>{£ +H<: +i )}< 



