58 BELL SYSTEM TECIIMCAL JOVRNAL 



integral of the impressed e.m.f. These principles are of considerable 

 practical importance in telegraphy. 



The leaky cable, that is, a cable with distributed leakage conductance 

 G in addition to resistance R and capacity C, is of some interest. The 

 differential equations of the problem are given in equations (70) ; the 

 operational formulas for the case of voltage directly impressed on the 

 terminals of the infinitely long line are 



1= ^^+ £g--Wc^^+^ Vo. 



Writing CRx^ = a. and RGx^ = P, G/C = \, and assuming a "unit e.m.f." 

 impressed on the cable, this becomes 



Y^g-^^f^^ (175) 



/= J^V^+X e-Va7+^. (176) 



These equations are readily solved by means of the table and formulas 

 given in a preceding chapter. 



But first let us attempt to solve the operational equation (175) for 

 the voltage by Heaviside methods, guided by the solution of the 

 operational equation 



V=e-^'^ (124) 



of the preceding chapter. Expand the exponential function in (175) 

 in the usual power series; it is 



F= 1 - v'aT+^+ (^^> - (j±tity^PS+ . . (177) 



Now discard the integral terms and write 



V=l- |l+-ir + -^V! +••• |Va/'+/3. (178) 



We have now to interpret the expression s/ap+lS. We have by 

 ordinary algebra 



/ /^ \ 1/2 / Xx 1/2 



V ap/ \ p/ (lyc)) 



=b+tp-UTp)'^^^{jp)' -^ ■ ■ -^^^p- 



