68 BELL SYSTEM TECHNICAL JOURNAL 



(3) In (m.2) make the substitution p= — q and h=^t, and ulti- 

 mately replace q b^^ p and ti by /; we get 



where Xi= — X and mi= ^ M- (They are, of course, as yet, arbitrary 



X X 



parameters, except that they are restricted to positive values). 



(4) Now if we compare (m.3) with the integral equation (208) 

 for F, we see that they are identical provided we get 



Mi = P. 



'\i = ia = a\/—l, 



which is possible, since p>(t. 



Introducing these relations, we have 



Here lo denotes the Bessel function of imaginary argument; thus 

 Jo{iz)=loiz). 



It follows from (m.4) and the integral equation (208) that 



F(/)=Ofor/<x/z;, (209) 



= e-'"Io((rV^'-^V^') for t^xjv. 



Having now solved for F=F{t), the current and voltage are gotten 

 from equations (206) and (207). Thus 



7 = for t<x/v, 

 = J ^ F{t) +vG j F{t)dt for / ^ x/v. 



The corresponding voltage formula is 



F=Ofor t<x/v, 



, — (211) 



Here 7i((t \/t'^ — x^/v'^) is the Bessel function of order 1 : thus -iJi{iz) = 

 Ii(z). The function is entirely real. The derivation of formula 

 (211) is a little troublesome, owing to the discontinuous character of 

 the function F: the detailed steps are given in an appendix. 



