268 BELL SYSTEM TECHNICAL JOURNAL 



the core-sheath circuit is neglected in comparison with propagation along 

 the sheath-earth circuit, which is permissible. The voltage between sheath 

 and core conductor differs by the factor z/Z from the voltage between 

 sheath and ground as given in Table II, case 3 of the paper referred to, 

 z being defined as before and Z being the unit length self-impedance of the 

 sheath-earth circuit. At a point opposite the lightning stroke, .r = 0, 

 the voltage between core and sheath is in this case given by: 



U{0, y) = J^ f Qo(r)e-'- dx =jI^ *(ry) {3,3,) 



Z Jo Z ZTT 



where Z = T~/G and G is the unit length leakance of the sheath-earth cir- 

 cuit. The leakance is given by the approximate expression: 



Vtt '' Taj 



G=(^^Jog-j (34) 



a being the radius of the sheath and log = log^. 



The function $(ry) is given by the approximate formula: 



HTy) = log ^^ (35) 



Inserting (34) and (35) in {ii), the latter expression may be written: 



r(0, y) = r(0, a)\{Ty) (36) 



where V{Q, a) = r(0) is the voltages when the current enters the sheath 

 directly (y = a) and: 



X(ry) = (^log ~^'')/iog i/ra (37) 



where V = {iwv/lp)' — (/coa)"'. 



The rigorous solution of the time function corresponding to (36) would 

 be rather complicated. Since, however, X is the ratio of two functions, 

 each of which varies logarithmically with T, and thus varies only slightly 

 with ico, an approximate solution is obtained by replacing /co with \/t in 

 (37). For instance, the solution of an operational expression p~" is t"/nl 

 while the solution of p~" log p is [\[/(\ -|- n) -f log l/t]t"/nl, \p being the 

 logarithmic derivative of the gamma function. For representative values 

 of n and t{n < 1, / < 10"'*), xp is less than 5% of log 1//, so that a good 

 approximation is obtained by replacing p by 1// in log p, which in this 

 illustration simulates the factor X(ry). With this approximation: 



r(0,y,0 -^ r(0, a, /)XLv(«//)^] (38) 



