Electric Waves round a Large Sphere. 525 



When m and z are nearly equal, 



^ = _ 7r (_i)24V3t7r^+ ... 



= — 2/\/3 when m = z. 



Thus at the critical point m = z, ^ n =^7r. This value 

 exhibits no discontinuity on passing the point. When m is 

 much greater than z, 



2T n =z/(m*~z*)i, 



2T n '=m 2 /{m 2 -z 2 )l, 



and by a comparison of substitutions for the Bessel functions, 



R„=2T n cosh2* n 



~ = 2TJ cosh 2t n + 2 sinh 2t w 

 OZ 



since by an identical relation, 



2TnU' = l (28) 



But t n is large and negative, and c)R„/dz is therefore very 

 great for a great value of n. Accordingly, %n tends to the 

 value \ir when n is infinite. 



Thus as n increases from zero to infinity, % n is never of 

 the order of <j> n in z. It ranges from zero to ^7r, taking the 

 value ^7T when n+\-=.z« Within the range of application 

 of the first type of expansion of the Bessel functions, it is 

 very small and of order z~ l .\ 



Harmonic terms of infinite order. 



When n is very great, a substitution ?i + i =m = z cosh yS 

 may be made, where z denotes ka as before. Thus from the 

 appropriate expansions of the Bessel functions, 



2Tn=cosech/3 -> 



*»=-£ log 2 +*(sinh £-£ cosh 0),/ * (29 ) 



a quantity ultimately very large and negative. 



Comparing the two types of expansion of the Bessel 

 functions, 



tan</>n = e 2 ' rt . 



