ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 541 



The connection between electromagnetic theory and line theory is 

 so important that, risking repetition, we wish to emphasize their 

 intimate relationship by deriving the well-known differential equations 

 of the line theory directly from the electromagnetic equations (2) 

 combined with the assumption that the longitudinal electromotive intensity 

 vanishes everywhere. We already know that under the assumed con- 

 ditions the first equation of the system (2) becomes 



where / is the total current flowing in the inner cylinder through a 

 particular cross-section and is some function ^" of s. We can therefore 

 rewrite the last two equations of the system as follows : 



dEp icofj. ^ I dl . 



oZ LTrp Airp oz 



We have merely to integrate both equations with respect to p from 

 b to a and substitute the potential difference V for the integral of the 

 transverse electromotive intensity to obtain 



dV / icofx a\ dl Iwicoe 



a^= -Ur^^^^j^' Tz=-T-a:^' (29) 



log^ 



which are the equations of the transmission line whose distributed 

 series inductance equals (fxllir) log (a/b) henries/cm. and shunt capacity 

 27re/(log a/b) farads/cm. 



With this, we conclude the special case in which the longitudinal 

 electromotive intensity vanishes everywhere, the propagation constant 

 equals icoVe^t, and the velocity of transmission is that of light. 



We now turn our attention to the case in which A and B do not 

 vanish. We have already noted that the propagation constants are 

 given by equation (20). Since, in this case, we are interested primarily 

 in the nature of the phenomena rather than in the details of field 

 distribution, we shall simplify our mathematics by supposing the 

 radii of the cylinders to be infinite. Thus, the cylinders become two 

 planes perpendicular to the x-axis, distance a apart. The 99-direction, 

 then, coincides with the 3;-direction and, therefore, all the intensities 

 are independent of the 3;-coordinate. Let us choose the z-axis half- 

 way between the planes. The equations describing this two-dimen- 



" On this occasion, we should remember that a particular type of this function 

 had not yet been ascertained at the time the equations (2) were arrived at. 



