Waves along Cylindrical Conductors of any Section. 203 



that there is no need for any such restriction, the manner 

 and velocity of propagation along the length being the same 

 whatever may be the character of the cross-seution of the 

 system. 



The form of <£, and the self-induction of the system, may 

 be determined in this case, whatever may be the radii (a 1} a 2 ) 

 of the wires and the distance (b) between their centres. If 

 r l3 r 2 are the distances of any point P in the plane from fixed 

 points Ox, 2 , the equipotential curves for which <j>, equal to 

 log (?y>'i), assumes constant values are a system of circles, 

 two of which can be identified with the boundaries of the 

 conductors. The details of the investigation, consisting 

 mainly of the geometrical relations between the ultimate 

 points Oi, 2 and the circles of radii a 1} a 2 , are here passed 

 over. The result for the self-induction per unit length L, or 

 for the capacity, may be written * 



1 2 2 2 / < / 1 2 2 2\2 a 2 2 ) 



T i) ,b—a l —a 2 —V\{b~a l — a 2 )—^a l a 2 \ 

 L=-21og ^ • ( 17 > 



As was to be expected, L vanishes when b=a 1 + a 2 , that is, 

 when the conductors are just in contact. 



When a x , a 2 are small in comparison with b, the approximate 

 value is 



L=-21«g^(l+^); .... (18) 

 or, if a 1 = a 2 =a, 



l = 4 K-p) ( i9 > 



The first term of (19) is the value usually given. The 

 same expression represents the reciprocal of the capacity 

 of the system per unit length. 



In the application of Lecher's arrangement to the investi- 

 gation of refractive indices, we have to consider the effect of 

 a variation of the dielectric occurring at planes for which z is 

 constant. It will be seen that no new difficulty arises in the 

 case of systems for which the appropriate function <f> in two 

 dimensions can be assigned. 



Regarding (f> as a given function, e. g. log r for the case of 

 a coaxal wire and sheath (compare (11)), we may take as the 



* Compare Macdonald, Cainb. Phil. Trans, vol. xv. p. 303 (1894). 



