ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 143 



further by using the differential equation satisfied by z~ 1/l (v m), which is BESSEL'S 

 equation.* 



The various approximations are collected together below for convenience of 

 reference : 



When zn\ is great compared with z 1/3 , 



R = sec a, (j) = z cos a ^mr + (n + ^) a (ii.), 



where z sin a = n + 1-. 



When zn\ is of the same order as z' : '\ 



When z n^l is of lower order than Z'\ 



V] . . . (iv.), 



= Jir- 3/xc (1 M sin ........ (v.), 



........ (vi.), 



where 



When n + ^z is of the same order as z \ 



where 



E = 6- 1 "2 I >~ I/ '[^ i +ie~ v l, tan^ = ^- vs ..... (viii.), 



3 



tan ^ = e 4M " .......... (i- x -)- 



When n + ^z is great compared with z' 3 



R = (sinhS)- 1 [e- 2r + ic 2r ], tan^ = ^ ...... (x.), 



T = z sinh S (n + ^) 8, 2 cosh S = n + ^. 

 When n is not an integer the corresponding results can be obtained by writing 



* Another method of approximating to the value of R is to make use of the relation 



E = f Ko(2 sinh f) cosh(2+ 1) {<#, 

 T Jo 



which is not difficult to establish, and then deduce <#> from the result ; the method given is more direct, 

 and avoids the difficulties that arise in determining the constants for the different forms of < in this other 

 method. 



