I9I0.] LONG ELECTRIC WAVES ALONG WIRES. 365 



coils k and k -\^ i is F/,-. For the first problem we then have 

 Fo= F„ = o. Let L^ be the coefficient of mutual induction between 

 any coil and one of its nearest neighbors ; L, the coefficient of mutual 

 induction between the same coil and its next nearest neighbor but 

 one, and so on. Similarly, let S^ be the electric induction coefficient 

 between any condenser and one of its nearest neighbors ; S., the in- 

 duction coefficient between two alternate condensers, etc. We can 

 then write for the ^th coil : 



and for the ^th condenser : 



+ S,V,_, + S,V,,,+ ■■■) + KV^ 



(0 



(2) 



Now in the case of long electric waves the currents in any coil 

 and its near neighbors will be very nearly the same. The terms in 

 the series in (i) containing currents in distant coils become rela- 

 tively unimportant on account of the diminution of their coefficients. 

 In this special case it will therefore be legitimate to replace the series 

 by a single term and we can therefore write : 



K-..-V, = L'^a-^R'C,, (3) 



// 



in which L' may be termed the effective coefficient of self-induction 

 of any one of the coils. When we pass to the limit by increasing 

 indefinitely the number of coils, etc., and at the same time decreasing 

 indefinitely all the electric constants, the limiting value which the 

 product of L' by the number of coils in a unit length approaches will 

 be the self-induction per unit length of the uniform line. Equation 

 (2) modified in an analogous manner reduces to: 



C,-C,., = S''^j^V, + K'V, (4) 



