and Resistance of Straight Conductors. 385 



take 



\\B.wdxdy = ^ fil3 2 rdr, .... (6) 

 J «' Jo 



in which /3 represents the magnetic force, everywhere per- 

 pendicular to r. 



In the integration from to oo there are five regions to be 

 considered. In the first, from to a 2 , there is no magnetic 

 force. In the second, from a 2 to %, the magnetic force depends 

 upon the total current travelling through the strata which are 

 internal with respect to the point in question. In terms of 

 the total current C we have 



*-&<:-$ m 



The permeability is here supposed to be fi. 



In the third region, between the cylinders, the permeability 

 is fi , and the magnetic force is given by 



*-T • • < 8 > 



Within the second cylinder the permeability is fj/, and 



^-^ik-f) w 



In the fifth region, from a{ to co } 0=0. 



Effecting the integrations, as indicated in (6), we obtain 

 the value of 



jj H w dx dy, 



which gives 2TJI ; and (if L be the coefficient of self-induc- 

 tion) T = i LC 2 . The result is 



L 1 a 2 



T =2^ log- 



a 2 —a 2 \ 4 a^ — a 2 *a 2 ) 



V ("a/ 2 - 3a/ 2 , a/ 4 , a/ 1 , 1/tt 



Perhaps the most interesting application of the general 

 result is to trace the diminution of self-induction as the two 

 currents are brought into closer and closer proximity. Let 

 us suppose that the intervening space is reduced without 

 limit, so that a 2 / = a 1 . Suppose further that /u/=//-, and that 

 both conductors have the same sectional area 7rc 2 ; so that 



% 2 — « 2 2 = a i /2 ~~ a 2 2 = ° 2 ' 

 Phil. Mag. S. 5. Vol. 21. No. 132. May 1886. 2 E 



