Self-induction of Wires, 133 



and requiring us to substitute 



O^^ + irlog {a 2 /r) + %r 



for (sir)- 1 in the H 2 and y 2 formulae in (28). Then (6) is 

 nearly satisfied, and is quite satisfied if we change the last 

 term in the last expression to %r. But the other fundamental 

 is violated. 



Now in (27) take m=0, making —sl=fjt, 2 cp 2 , and bringing 

 (27) down to 



%8ia 1 J {s 1 a 1 ) = -^pJ 1 (s 1 a 1 ); . . . (29) 



where 



L =2^ 2 logfe/a 1 ), 



the coefficient of self-induction of the surface-current, and 



Ho=(tt^)-S 



the resistance of the wire, both per unit length of wire ; so 

 that L /R is the time-constant of the linear theory, on the 

 supposition that the resistance of the wire fully operates, 

 although the current is confined to the surface. This case of 

 m=0 is appropriate when the line is so short that the electro- 

 static induction is really negligible in its effects on the wire- 

 current. In fact we shall arrive at (29) from purely electro- 

 magnetic considerations, with c = everywhere. But it is 

 also the proper equation in the m = case when the electro- 

 static retardation is not negligible. It must be taken into 

 account, for instance, in the subsidence of an initially steady 

 current, independently of the electrostatic charge. 



Equation (22) in powers of p, by means of \s\a\— — /^/Rq 

 We get 



1 +«+*(«) , +....-^(i + a£ + ..:> (30) 



Taking first powers only, we get 



-p- 1 = (fx 1 -\-L )/U ; 



which is greater than the linear theory time-constant of the 

 wire by the amount J/*i/Rq, since J/^ is the coefficient of self- 

 induction per unit length of wire when the return-current is 

 upon its surface. 



But taking second powers as well, we get, if L=^/^ 1 + L , 



—p- 1 = L/B, and i/^i/R ; 



of which the first is exactly the linear-theory value. The real 

 time-constant of the first normal system of current, therefore, 

 exceeds the linear-theory value by an amount which is less 



