234] ELECTROMAGNETISM. 465 



where r 12 is the mean distance between points of the two cross- 

 sections. The second integral is 



where r u is the mean distance between pairs of points of the 

 section 8 lt and the eighth integral is the corresponding quantity 

 for the area 8 Z . Accordingly the self-inductance of the whole 

 circuit per unit of length is 



(21) j = p. (4 log r 12 - 2 log r u - 2 log r a ) = fy log ^_- . 



If the conductors are circular cylinders, we may use the formulae 

 already found. If both the direct and return conductors are 

 single cylinders external to each other, their axes being a distance 

 d apart, r 12 = d. For infinitely thin tubes of radii R and R', 

 r u = R } r<n = R', so that 



(22) -^ 



For tubes of radii R l} R 2 and R^, R 2 ', integrating (17) over the 

 area of the ring, 



E^ log R 2 - R 



- {Bf (log Bf -l)-Bf (log JV - 1)} + 



and making reductions 



^R 2 2 -^i 4 i -^i 



^) + (Bf - R?? log S 



From this we obtain r^ by replacing B^, R 2 by -R/, i^/, so that 

 we obtain for the self- inductance of the circuit 



, . L f. d 2 RS . # 2 ^ /4 . E/ 



(2 4 ) = V log + _ log + _ log 



For solid wires, since lim (R logR) = 0, this becomes 

 (25) T = 2 



w. E. 30 



