524 BELL SYSTEM TECHNICAL JOURNAL 



For values of ber x, bei x, ber' :x; and bei' x see Jahnke u. Emde, " Funk- 

 tionentafeln." M^"^ and M are given above in emu. per centimeter. 

 We assume M'-^'' and M proportional to the length and multiply by 

 0.1880 to obtain michrohenries per 74 inches. 



No assumptions with respect to frequency are made in formulas 

 (13) and (14) but terms of the order of magnitude with respect to 

 unity of 9y^i^ ~ 9X^ or smaller are neglected. That is, the accuracy of 

 (13) and (14) is limited by the dimensions rather than by the frequency. 

 But for frequencies of about 100 kilocycles or higher and not too small 

 wires (that is, when x ^ about 10) formulas (13) and (14) may be 

 expressed in interpretable form; namely, 



ilf (0) = lf„(o) -I- iMfaC) (15) 



and 

 where 



M„^o) = 4(log. V2 - ^-^, + 3-^, 



M = Ma + iMb, (16) 



X2 . X2 



Mb^o) = - 4 



X2 



1 - 4X 



M„=M.co)-4(^,-3Vj+4.^^^^,-6X^ 



M, = M,(o) - 4:u( Y^^2 - 6X^ 



The asymptotic values (when / or o- or both approach infinity) are, 

 therefore, 



M(«> = 4(log, V2 - ^ ^^2x0 ' 



(17) 



M= M(«) - 4 ^ ^'2X^ - 3X^^ (18) 



and the d.c. value (when / approaches zero) is, of course, 



ikT = 4 log, V2. 



Thus M is real (i.e., Mb = 0) when the frequency is either zero or 

 infinite. 



Formal Solution 



The following derivation of these results is an application of the 

 general method of calculating the self and mutual impedances in a 

 system of parallel wires which is outlined in Section V of John R. 

 Carson's paper "Rigorous and Approximate Theories of Wave Trans- 

 mission along Wires," B. S. T. J., Jan., 1928. This method of solution 



