742 BELL SYSTEM TECHNICAL JOURNAL 



long line, therefore, tlie operational formula for the current entering 

 the line is 



/=JIgLr„ (97) 



'■\pL + R^'" 



where 1', is the \-ohage at the hue terminals. If this is a "unit e.ni.f." 

 we have, as our operational equation, 



I = J P^ (OS) 



\ pL+R 



which can he written as 



IC 1 



/ = J^--=L== (99) 



^L Vl+2X'/> 



(100) 



where \ = R 2L. 



The corresponding integral equation is 



C 1 



From either equation (09) or (100) and formula (p) of the table of 

 integrals, we see at once that the solution is 



/ = ^|'c-^'/„(\0 (101) 



where /(,(X/) is the Bessel function 7„(/X/), where /=\/— I. (The 

 function is, however, a pure real.) 



Heaviside's procedure, in the absence of an>' correlation between 

 the operational equation and the infinite integral, was quite different. 

 Remarking, with reference to equation (00), that "the suggestion to 

 eni[)lo\- the binomial theorem is ob\ious," he expands it in the fortn 



^-N'l r p^2\[p) 3\ \p) ^ ■■ \ 



and replaces \/p" by t"/n in accordance with the rule discussed in 

 preceding sections. The e.xplicit solution is then 



a convergent solution in rising powers of /. As yet, howe\er, he does 

 not recognize this series as the power series expansion of (101), which 

 it is. He does, however, recognize the practical impossibility of 

 using it for coni|)uting for large values of /, and remarks "But the 

 binomial theorem furnishes another way of expanding the operator 



■=^l 



