280 Mr. 0. Heaviside on the 



wherein everything is the same as in Sir W. Thomson's system, 

 with the addition of the electric force of inertia — LpC, where 

 L is the coefficient of self-induction, or, as I now prefer to 

 call it, for brevity, the inductance, per unit length of the wire, 

 according to Maxwell's system, being numerically equal to 

 twice the energy, per unit length of wire, of the unit current 

 in the wire, uniformly distributed. Coming after Maxwell's 

 treatise, there is of course no question of any important step 

 in advance here, except perhaps in the clearing away of 

 hypotheses involved in KirchhofPs investigation. 



The system (71) is amply sufficient for all ordinary pur- 

 poses, with exceptions to be later mentioned. It applies to 

 short lines as well as to long ones ; whereas the omission of L, 

 reducing (71) to (70), renders the system quite inapplicable 

 to lines of moderate length, as the influence of S tends to 

 diminish as the line is shortened, relatively to that of L. An 

 easily made extension of (71) is to regard R as the sum of 

 the steady-flow resistances of wire and return, and V as the 

 quantity Q/S, Q being the charge per unit length of wire. 

 Nor are we, in this approximate system (71), obliged to have 

 the return equidistant from the wire. It may, for instance, 

 be the earth, or a parallel wire, with the corresponding changes 

 in the formulae for the electrostatic capacity and inductance. 



But there are extreme cases when (71) is not sufficient. 

 For example, an iron wire, unless very fine, by reason of its 

 high inductivity ; a very thick copper wire, by reason of 

 thickness and high conductivity ; or, a very close return 

 current, in which case, no matter how fine a wire may be, 

 there is extreme departure from uniformity of current dis- 

 tribution in the variable period ; or, extremely rapid reversals 

 of current, for, no matter what the conductors may be, by 

 sufficiently increasing the frequency we approximate to 

 surface conduction. 



We must then, in the system (71), with the extension of 

 meaning of R and V just mentioned, change R and L to R / 

 and I/, as in (67), and other equations. In a S.H. problem, 

 this simply changes R and L from certain constants to others, 

 depending on the frequency. But, in general, it would I 

 imagine be of no use developing R/ &c. in powers of p, so 

 that we must regard (R^ + L/p) &c. merely as a convenient 

 abbreviation for the R^ &c. defined by (56) and (55). 



A further refinement is to recognise the differences between 

 W and 1/ in one m system and another, instead of assuming 

 m = in R^. And lastly, to obtain a complete development, 

 and exact solutions of Maxwell's equations, so as to be able to 

 fully trace the transfer of energy from source to sink, fall 



