286 BELL SYSTEM TECHNICAL JOURNAL 



where 



Hy) = log (y^^) / log (i/ra) 



r being the propagation constant of the sheath-earth circuit and a the radius 

 of the sheath. The solution of the latter equation for y is : 



1 /2pA* 1 



y = y^(y ^ wT^v^"^^ = [~) /"-o/i _ 1 (90) 



where 



m = log (l/cF) 



r ^ {v/2pty per meter 



z^ = 1.256-10" henries per meter 



t = 10^'* sec. = time to crest of core-sheath voltage. 



When (89) and (90) are equated, the following expression is obtained: 



i^i/""" - 1) = (~y (91) 



The value of i which satisfies the latter equation is the value ii defined 

 above, and is shown in Fig. 9 as a function of mio for various values of 



When (89) and (90) are inserted in (88), the latter integral may be ex- 

 pressed as follows: 



n = 2Nsf^ (q[H{io) - H{h)] + rjS G(h, miA (92) 



where the current is in kiloamperes and: 



TV = Number of strokes to ground per square meter 



.y = Length of cable, in meters 



q ^ .08 when p < 100, q ^ .047 when p > 1000. 



H(i) = -[ PPo(l)di (93) 



G{l, mto) = ^mioll _ J - j mioli _ J (94) 



The first term in (94) equals YPo(I) = V2(/)-Po(/). Since Po(0 is nega- 

 tive, the above integrals will have positive values. 



The term q[H{io) — H(ii)] of (92) gives the portion of failures due to 

 direct strokes while the term involving the function G gives the portion of 

 failures due to ground strokes not necessarily arcing to the cable although 



