and Inductance of a Concentric Main. 



The equation now becomes 



d' 2 " 1 "du 2 n 



27 



(B) 



The solution of this equation is known * to be 

 m=A . I (^/'\Aj + B . Ko(mr^/T), 



where A and B are constants, and I () (/) and K//<:) can be 

 computed by means of the following series :— 



When x is .-mall. 



(2) 



(3) 



and 



where «=log 2—7 and 7 is Euler's constant, and so 



*== 0-1159315. 

 When .v is large. 



e* r, l 2 I s . 3 s 1 



J '"-v^l 1 + ^ + X^" I' ' 



and 



Wj^l^s+SJ" 1- (4) 



The values 0! the series (2) and (4), for many value- of *, 

 have been computed to a high degree of accuracy by A\ . 8. 

 Aldis t, and tables of l^x) from to 5'1 are given in the 

 British Assoc. Report-. 1896, the interval of the argument 

 beine 0*001. 



2. Kelvin's ber and bei junctions. 

 Kelvin % showed that the effective resistance of the inner 

 conductor of a concentric main may be conveniently expr 



in terms of two functions which he called the berand the bei 

 fnnctions§. He published tables of the values of the^e func- 

 tion- for a few values of the argument, and from these tables 



* See Grav and Matthews, ' Jie.-sel's Functions.' 



t Proc. Roy. Sue. vol. lxiv. p. 203. 



X L. c. ante. . _ . , 



§ Noticing that in Heaviside's notation, ber = M and bei = > , Kelvin a 

 formula follow, at once from the formula (36) given in Heavi-:dv a 

 ' Electrical Papers/ vol. ii. p. 183. 



