Forms of the Lines of Electric Force. 311 



The diagram thus gives a distorted picture of the arrange- 

 ment of the lines in the two opposite extreme cases referred 

 to above. The broken lines represent the lines of force and 

 the full lines those of Poynting flux. In the eddy region 

 from 135° to 180° the arrows on the energy lines are taken 

 in or out according as we are dealing with a case of " l\ 2 a " 

 small or large, as already explained. 



To pass to the actual arrangement in experimental con- 

 ditions one has to imagine the sag of the lines of force re- 

 duced and the region of re-entrant lines to shrink to a very 

 small length at 135°. So that except in the immediate 

 neighbourhood of this point, where the radial force vanishes, 

 the lines go practically straight across. 



§ 6. Forms of Internal Lines of Force and Flow. 



To the degree of approximation adopted in this paper, 

 these are the same as for the case of a single wire, the return 

 current being carried by the dielectric (Sommerfeld's case). 

 When there is axial symmetry in the field we can, as Hertz 

 showed *, express the quantities Z and R in the forms 



r or\ o?' ' 



13/ a^ 



tqz\ or J 



so that the lines of force are given by 



a* 7 



r.sr— = const. 

 Or 



Using this method Sommerfeld (he. cit. p. 285) has given 

 a diagram showing the course of the internal and external 

 lines of force, with distortion, for the case of surface con- 

 duction {k 2 a large). He does not draw attention to the back- 

 ward tilted lines and, to judge from his figure, seems not to 

 have plotted lines in this region. 



As I wish to get the lines of energy-flow also, I shall 

 proceed directly from the differential equation, taking 

 separately the cases of (a) k 2 a large, (b) h 2 a small. 



(a) In this case the conduction is confined to a small 

 thickness at the surface of the wire, so that we may use for 

 Jo(l' 2 r) and Ji(A- 2 r) occurring in Z, R, the forms appropriate 

 to large values of the argument. 



* Hertz, l Electric Waves ' (English translation), p. 140. 



