220 Prof. 0. J. Lodge on the Theory 



of strain in it. If the whole is being discharged, then the 

 initial state of strain is not uniform, while the final is.) 



Calling the whole current down the axis C , we have an 

 equal inverse displacement all over the area 7r(b 2 — a 2 ), so 

 that the density of its distribution is cr, where C =7r(6 2 — a 2 ) a: 



The intensity of magnetic force at any distance r from the 



axisis ^,_ 2(C -C) 



/ - r > 



where C is that portion of the displacement recovery which 

 lies nearer to the axis than r. This is accurate, for the dis- 

 tribution of the current matters nothing so long as it is in 

 coaxial cylinders ; the portions external to r have no effect. 

 On the hypothesis of uniform distribution, 



C = ^r*-a 2 )*=^ 2 C . 



Hence 2Co b 2 -^ 



J r 'b 2 -a 2 ' 

 where the a may in practice be neglected as usually too small 

 to matter. This is the number of lines of force through unit 

 area at the place considered ; and the whole magnetic induc- 

 tion in the cylindrical space considered outside the conductor is 



aT fifdr=fjLhG (2 \og h - -l\ 



For the part inside the conductor there is an extra term to 

 be added, which, on the hypothesis of uniform distribution in 

 the conductor, comes out 



and which may really have any value between this and zero, 

 according to the rapidity of the alternations, and the consequent 

 deviation from uniform distribution. 



The entire magnetic induction may be written LC ; hence 

 we get the value of L, the coefficient of self-induction, or the 

 inductance, of the circuity 



L = A(Vlog--/* + ^o) C 1 ) 



This I shall write for convenience h(/jbu 2 + fi ), so that u 2 is an 



b 2 

 abbreviation for log —% — 1*. 



* It is quite likely that my calculation of this term u is faulty. But 

 the truth of what follows is not affected by such an error, and a 

 mathematician will be easily able to set it right. 



