and Inductance of a Concentric Main. 



537 



When x is not small we get suitable formulae for calculating 

 these functions by noticing that 



= — lVx / L + ciq + h ' y/o 



is also an approximate solution of (17). Hence, finding 

 a series in descending powers o£ x for 6, and determining 

 the constants by (4), we find that 



and 

 where 



and 



ker a? = ^/ ~e a cos ft', . 

 25 



kei x = 



1 



13 



s/2 SV2x 384 \/2x s 128 x 



i'+ 



ff = 



' -?+ 



1 



+ 



25 



V2 8 %V2x 16\v 2 384 v 2a; 3 



. (42) 



. (43) 



. (44) 



-.... (45) 



It will be seen that a! and /3' can be deduced from the 

 formulae for a and /3, (18) and (19), by merely changing the 

 sign of x in the latter. In making calculations it is best to 

 use the formulae obtained by writing — x for x on the right- 

 hand side of the equations (22) and (23). 



By differentiating (42) and (43) we find that 



ker 



and 



'.v= — ker#<— /='+« =- > 



V2 2* 8i/2a? 2 j 



, . f 1 1 1 ) 



(x/2 8 \/2a; 2 8.r 3 /' 



— keia?<- -- +h ; — »>• 



(x/2 2ctJ 8V2.U 2 ) 



(46) 



kef\i-= - 



We also have 



7T6 



2a' 



X^.i') = ker 2 x + kei 2 a? = — - 



2x 



(47) 



(48) 



and 



YlW=k e^. + kei^=X 1 (,){i + -i- 2 f-^-^- 5 }.( 4 9) 



