Forms of the Lines of Electric Force. 309 



(from to 7r) between points at which the lengthwise electric 

 force Z vanishes, we find the flow of energy 



Outside. 



from to (tt — a) forward, in 



from (77 — a) to {tt— /3) backward, in J- . . (13) 



from [it— /3) to ir forward, out 



Inside. 



from to I 3- — a \ backward, in 



1 



from (- — a\ to (V— /3) forward, in /. (14) 

 from [it— /3) to ir backward, out ; 



The extent of the eddy in the external energy-flow is given 

 by a. Its course as shown by the curve is in accordance with 

 the statement in § 1. For the limiting case of mere surface- 

 conduction (k 2 a large) a = /3 = 45°, the section (77-— a) to 

 (77-— /3) shrinks to nothing, i. e. the energy-How is everywhere 

 in the direction of propagation of the waves. At the other 

 extreme (k 2 a small) when a again approaches 15°, j3 becomes 

 zero, and the region of outward flow disappears. 



§ 5. Forms of External Lines of Force and Flow. 

 We may write for points between the wires 



ry i l> — T 7 • 27T,S 



Z = log e~ kz sin ~^— 



11 A 7 . f^irz \ 



(15) 



where »?& 



A= -s- 



The differential equation to the lines of electric force is 

 therefore n 



• r 27r . \ 



dr A <m T~~ + a 



7 = ^-— ._A^ L, . . (16) 



r(b-r) log—- sin-— 



* A, 



and the integral of this 



A\ f 27T2; . . . 27rz -} 



2^| C0S a ' a7 + sm a lo £ * ln X J 



=C- H(* + ^)(ft-* , ) f log 4 (5-r}+ (3b-2ry log r+br(b-r)}, 



