MUTUAL IMPEDANCE OF GROUNDED WIRES 



265 



M{r,II,li) = Mu{r) + Mi{r,II + h) - M^{r,\n - h\), 



Po{r) = 



2irr 



47r Jo I /I M^ 





}-/o( 



riJL)dfx, 



P2{f',d) = 4— M^ log -^ ('' + " ) + ^ 



J\hir) =2^3[1 - (1 + I»g-'>], 



M,{r,s) = '-f r (1 - .--) 

 47r Jo 



1 



(^2 ^ r^)!/'' - M 



Jo(rn)dij., 



M2{r,d) 



_ icoi' r 1 



47r [ ^ 



r (r^ + d^)'i 



The integrations in the iterated integral are extended over the two 

 wires 5 and s, lying in planes at heights // and h, respectively. The 

 elements dS and ds are separated by the horizontal distance r and 

 include the angle e between their directions. The propagation con- 

 stant of plane electromagnetic waves in the earth, varying with the 

 time as e'"', is r, which equals {io^v/pY'^. All distances are measured 

 in meters, Zu in ohms and p in meter-ohms; v has the value 1.256 

 X 10"^ henries per meter; to is equal to 27r times the frequency; Jo is 

 the Bessel function of order zero. The derivation of the formula is 

 outlined in the latter part of this paper. 



The functions P and M are divided into three parts: first, Po and 

 Mo, which are functions only of the horizontal distance r; secondly, 

 Pi and Ml, which are functions of r and of the sum of the two heights 

 // and h ; and thirdly, P2 and if 2, which are functions of r and of the 

 numerical difference of the two heights. These three parts are 

 arranged in the order of relative importance when the heights are 

 reasonably small. For zero heights, the functions P and M reduce to 

 Po and Mo, which are the values previously obtained for wires on the 

 surface. For small values of the heights. Pi and Mi are of the order 

 of magnitude of the sum of the heights, whereas P2 and il/2 are of the 

 order of magnitude of the square of the difference. 



For some purposes it is convenient to transform formula (A) into 

 the alternative expression : 



Z12 



-//[ 



d^P(r,n,h) 

 dSds 



+ cos e M{r,II,h) 



dSds, 



(B) 



