TRANSIENTS IN GROUNDED U'lRES 431 



The first form of ^'(^0 may be checked directly from equation (3) 

 by introducing ico = p in the operationally equivalent function of p; 

 the third term of (3) being expressed by the infinite integral: 



Fip) = p e-i"f{l)ilt. 

 Jo 



The second form of "^(u) is obtained by separating the d.-c. mutual 

 resistance term, and transforming the infinite integral as follows: 

 express the error function in integral form, put y — 7z;/(2Vw) where 

 y is the variable of integration for the error function, and invert the 

 order of integration; thus 



I exp — w — - — ei I — -;= aw 



V^ Jo ' Jo ' \ '' 4w' / Vzl- 



exp ( - 7 Vx- + v^)dv. 







The infinite integral evaluated in the third line is No. 495 in Peirce's 

 " Short Table of Integrals," third edition. 



The second form of ^(«) may be verified by direct double integra- 

 tion of the mutual impedance; it agrees with the known result in the 

 limit for one wire infinite, and, when expanded in powers of y, with 

 the terms given in the second form for the mutual impedance by R. M. 

 Foster, loc. cit. 



Expressions for voltages due to suddenly applied currents 

 exp (— kt) sin co/ or 1 — exp (— kt), which are important forms for 

 a.-c. and d.-c. networks, may be readily obtained from equation (4), 

 the first by use of the expression: 



exp (— kt) sin cot = y. [exp {— kt -\- it^t) + exp {— kt — iwt)~\ 



and the second by the substitution — ^ = /co and subtraction from 

 the unit step voltage. 



The results attained in this paper depend in appreciable measure 

 on advice and suggestions received from Mr. R. M. Foster of the 

 American Telephone and Telegraph Company; I am also appreciative 

 of the interest and advice of Messrs. K. L. Maurer and H. M. True- 

 blood of this company. 



