Waves along Cylindrical Conductors of any Section. 20 L 



The problem is accordingly reduced to dependence upon 

 a simple potential problem in two dimensions. Throughout 

 the dielectric </> satisfies 



d 2 <j>ldx 2 + d 2 (f>/dy 2 = (7) 



At the boundary of a conductor, supposed to be perfect, the 

 condition is that the electromotive intensity be entirety 

 normal. So far as regards the component parallelto z this 

 is satisfied already, since R = throughout. The remaining 

 condition is that be constant over the contour of any con- 

 tinuous conductor. This condition secures also that the 

 magnetic induction shall be exclusively tangential. 



It is to be observed that R is not equal to dcf)/dz. The 

 former quantity vanishes throughout, while d(f>/dz remains 

 finite, since <j> oc e i(pt+mz) m Inasmuch as (f> satisfies Laplace's 

 equation in two dimensions, but not in three, it will be con- 

 venient to use language applicable to two dimensions, referring 

 the conductors to their sections by the plane xy. 



If a boundary of a conductor be in the form of a closed 

 curve, the included dielectric is incapable of any vibration of 

 the kind now under consideration. For a function satisfying 

 (7) and retaining a constant value over a closed contour 

 cannot deviate from that value in the interior. Thus the 

 derivatives of $ vanish, and there is no disturbance. The 

 question of dielectric vibrations within closed tubes, when m 

 is not limited to equality with p/V, was considered in a former 

 paper*. 



For the case of a dielectric bounded by two planes perpen- 

 dicular to x we may take 



( p = xe i(pt + mz) } (g) 



giving 



p-^+w), Q = o, R = 0, . . . (9) 



a=0, &=_V-V<**+ W *>, c = 0, . (10) 



in which, as usual, m=p/Y. Since Q = 0, R = throughout, 

 the dielectric may be regarded as limited by conductors at 

 any planes (perpendicular to x) that may be desired. 



If the dielectric be bounded by conductors in the form of 

 coaxal circular cylinders, we have the familiar wire with 

 sheath return, first, I believe, considered on the basis of these 

 equations by Mr. Heaviside. We may take, with omission of 

 a constant addition to log r which has here no significance, 



<£= logr.e i( P t+mz) , (11) 



* Phil. Mag. vol. xliii, p. 125 (1807). 

 Phil. Mag. S. 5. Vol. U. No. 267. August 1897. F 



