Wave Propagation over Parallel Wires. 619 



Equations (34) constitute an infinite system of equations 

 in the infinitely many variables q 1 . . . q n , and on their 

 solution depends the determination of the harmonic co- 

 efficients h 1 . . . h n . 



The solution of (34) is to be obtained by some process 

 of successive approximation. For example, a formal solution 

 is gotten by taking q^. . . q n as the limit of the sequences : 



qi m , ?i«>, gP, ?i (3) , • • • ?i (s) • • • 



(0) n (1) n (2) n (3) n (s) 



i > Hn J 9n ' Hn » * ' * Hn > 



where the successive terms of the sequences are defined by 

 the relations : 



q y={-\yip n k<\ « = 1,2,3... 



aDd ?„ (s+ " = (-l)"2p n ^-^ l ~pjf> S„ ( 2 «). ■ 



The method of solution results in a convergent sequence 

 provided the parameter k is less than its limiting value 1/2, 

 and for values of k likely to be encountered in practice 

 a very rapidly convergent sequence. 



Another method of successive approximations which may 

 often be advantageously employed may be termed the method 

 of successive ignorations. This consists in first ignoring all 

 the variables except q 1 and determining its first approximate 

 value —2p^ from the first equation of the system. A second 

 and higher approximation is then gotten by retaining ^j and q 2 

 and evaluating them from the first two equations. A third and 

 still higher approximation results from retaining q Xy q 2 , and g 3 

 and solving for them from the first three equations. This 

 process is to be continued until the convergence of the 

 sequence is evident. This latter method of solution likewise 

 results in a convergent sequence, and works very well in 

 practice unless the parameter is too close to its limiting 

 value. 



While some such process of approximation is to be 

 employed in the general case, and indeed has been success- 

 fully applied by the writer to several similar problems, 

 a simpler method of solution fortunately suggests itself 

 in the special case of greatest practical importance — namely, 

 when the wires are composed of non-magnetic metals 

 and, in consequence, the permeability /x is equal to unity. 

 The resulting formulae have the added advantage of being 



