ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 539 

 the following equation: 



A Yo(mb) Y(,(ma) 



B Jo{mb) Joima) 



(18) 



This equation has an infinite number of roots * whose approximate 

 values can be readily determined if we replace Bessel functions by 

 their approximations in terms of circular functions. Thus, we have 



«7n = -^, (w = 1,2, 3, •••). (19) 



This is a surprisingly good approximation for all roots if the radius of 

 the outer conductor is less than three times that of the inner; and the 

 larger the n, the better the approximation.^ The propagation con- 

 stants are computed from the corresponding values of w„ by means 

 of the following equation, 



r„ = \m„2 - co^e/x. (20) 



First of all, let us study the simplest solution in which both A and 

 B vanish. In this case, the longitudinal electromotive intensity 

 vanishes identically. The magnetomotive intensity — and the trans- 

 verse electromotive intensity, as well — also vanishes unless the de- 

 nominator m^ in equation (14) equals zero. If all intensities were to 

 vanish, we should have no field and there would be nothing to talk 

 about; hence, we take the other alternative and let 



r2 + cahti = 0, i.e., r = icosliT, (21) 



the positive sign having been implied in writing equations (13). In 

 air, e/i = (l/c^) where c is the velocity of light in cm.; hence, in air 

 this particular propagation constant equals iu/c. Since E^ equals 

 zero everywhere, the electromotive intensity is wholly transverse; and 

 the flow of energy being, according to Poynting, at right angles to the 

 electromotive and magnetomotive intensities, the energy transfer is 

 wholly longitudinal. 



The above method of determining the propagation constant may 

 be open to suspicion ; besides, the method does not tell how to obtain 

 the actual values of the electromagnetic intensities but merely leads 

 to a relation compatible with the existence of such intensities. There- 

 fore, let us obtain the wanted information directly from the funda- 



* A. Gray and G. B. Mathews, "A Treatise on Bessel Functions" (1922), p. 261. 

 ^ It is strictly accurate if the radii of the cylinders are infinite, i.e., if we are dealing 

 with a dielectric slab bounded by perfectly conducting planes. 



