620 Mr. J. R. Carson on 



asymptotic in character, and consequently give the values 

 of q l . . .q n with increasing precision in the practically 

 important range of values. It should be remarked that 

 the formulae now to be derived constitute asymptotic 

 solutions also when//, is greater than unity ; they cannot, 

 however, be safely applied when the permeability is large 

 unless the frequency is very high. 



Restricting attention, therefore, to the case where fi=l, 

 we observe that the functions p n and cr n of (32) may be 

 written as 



Pn=-J n + l! J n-r 



These identities follow from the definitions of equations (32) 

 and well-known properties of Bessel functions. We know 

 also that when the argument f is large compared with 

 the order n, the function p n becomes closely equal to its 

 limiting value unity. We are therefore led to consider 

 the auxiliary system of equations in the auxiliary variables 

 Pi — .p n , which is obtained from (34) by replacing the 

 functions p x . . . p n therein by their common limit unity. 

 That is, we define the auxiliary variables P\>--p n by the 

 following system of equations : — 



p n=[ -i)«n>.-i-2-y^z n (p), . . . (35) 



01 = 1,2, 3,...)- 



Now, since the functions p 1 ...p n approach the common 

 limit unity as the argument f approaches infinity, it is 

 evident that p x . . ,p n are simply the limiting values assumed 

 by the variables q 1 . . . q n when the wire is of infinite con- 

 ductivity. From the known surface distribution of magnetic 

 force in this case the following solution of equations (35) 

 at once suggests itself, and may be readily verified; the 

 variables p 1 . . . p n are simply the Fourier coefficients of the 

 expansion 



l + 2ftcosfl = K(1 +Pl C ° S ° + p2 cos26+ '" ')' ( 36 > 



From (36) it is easy to show that 



Pi=-2h, i 



Pn= (-1)»2*V, J 



(37) 



