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BELL SYSTEM TECHNICAL JOURNAL 



As before, u is to be replaced in the equation by tiie functional 

 arguments, which are the four sums of the s-coordinates of position. 

 The factor x in \p{u) is introduced to make it a function of two 

 parameters, ux~^ and ttXx^/"'; the result of integration is x~^\}/{u). 

 The result has the dimensions of abohms when all quantities are in 

 electromagnetic c.g.s. units. 



To check equation (3) notice that the integration of the first term 

 of equation (2) is effected by removal of differentiation and integra- 

 tion signs, and substitution of limits; its contribution is identical with 

 the d.-c. mutual resistance.® The integration of 0(w) may be effected 

 by integrating the first term by parts and employing the indefinite 

 integral: 



erf (ax)dx = x erf (ax) H pcxp (— a-x"^) -f const. 



/' 





The result is checked by differentiating, that is, by the relation: 



du 



X ^\p{ii) + 



1 



V.v^ + it^ - 



For large values of ii, 



1 — exp 



= 0(")- 



rX.r^ 



/ 



smce 



erf (± =^) = ± 1, 

 so that for a = co the unit step voltage approaches the limit: 



7rX.%'" 



VM = 



7rX.V- 

 / 

 ttXx^ 



1 — exp 



1 - exp - 



/ 



7rX.r^ 

 t 



where / = S2 — -i is the length of the second wire. 



This result is in agreement with a result published by F. Ollendorff, 

 Elektrische Nachrichten — Technik, October, 1930, eq. (26), and by L. C. 

 Peterson, Bell System Technical Journal, October, 1930, equation (5). 



The case of collinear straight wires is obtained by taking the limit 

 ^c = 0, which gives 



lim X ^\p(7i) = - 



X=Q u 



- 1 + 



/I 



V2 



ttXh- 

 t 



erf 



ttX 



U A 



\ t 



w-'f(«). 



exp 



r)] 



This result involves the evaluation of an indeterminate form. 



^ G. A. Campbell: "Mutual Impedances of Grounded Circuits," Bell System 

 Technical Journal, October, 192.^, eq. (3), p. 5. 



