608 Prof. W. B. Morton on some Cases of Propagation 

 .'. the equation for c is 



Kj^ca) k 2 3j{k 2 a)' * *' 



Now ca is always, as Sommerfeld has shown, a very small 

 quantity (order 10~ 4 or 10~ 3 ) ; K ft , K x therefore approximate 

 to the values 



K = log — ] 

 to 7ra I 



ca 



where 7 is the constant T781 . . 

 The equation then reduces to 



>, ...... (7) 



J 



If we write «■£ for ^- this assumes the form 



x log # = constant =y, say. ... . (9) 



Sommerfeld has shown how a numerical solution of this 

 transcendental equation may be found by an ingenious 

 method of successive approximations, and has worked out a 

 number of typical cases. He has proved that we may write 

 the result in the form w= — %y, where is a quantity which 

 varies very slowly with ?/. Sommerf'eld's typical examples- 

 cover the whole range of practically interesting cases, and for 

 them B comes out a magnitude with a negligible imaginary 

 part, and lying between -^ and 3L-. We shall have occasion 

 to employ this approximate solution, using © to mean a real 

 fraction of this order of magnitude. It is usual and con- 

 venient for purposes of discussion to distinguish two extreme 

 cases, viz., those in which k 2 a, the argument of the J functions, 

 is very large, and those in which it is very small. The 

 former condition is secured by low resistance and perme- 

 ability, high frequency and. large radius ; reversal of any of 

 these circumstances tends to bring a case under the second 

 head. It will be seen that large (Jc 2 a), favours the development 

 of. the " skin effect.'"' The corresponding values of the con- 

 stant f are 



large (k 2 a) f= j±- 



small (#2«) / = 



I 



