ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 135 



Hence 



v-ai = 2-^TT- l! >z / -f\-i\\- i:shl '+ (a ^' l de + r e -- sinh * + <" +l ->*rty 



L O ^ 



-t, sin (tt + |) ir\ K --si>l<-<>'-fY 2 H , 

 Jo 



It- is required to find approximations for the integrals above when z is large. Taking 

 first the case where n + ^ is less than z, the important part of r iu arises from the 

 first integral on the right-hand side, and the most important part of that integral is 

 contributed by the values of 6 for which the exponent is nearly stationary ; writing 



iv = z sin 0+ (n + jf) 0, 



w is stationary when z cos 6 = n+i, and putting n + ^ z sin a, the corresponding 

 value of 6 is JTT a. Hence substituting # = I-TT- a-f .!), 



/ 1\ /I X 1 



w = 2 cos a+ (n-t-A) (ij-Tr a) +*2 cos a H s sin owr + , &c., 



7 ^ 3 ! 



and 



that is 



|" e -,.- sin + ( + '/,), ^0 = g- cos +(+ (V^-a). p "giAacoprf^+iAaBlnrtH....^ 



JO J >->/!* 



Now unless ^TT a is small, the important part of the integral on the right-hand 

 side is 



_ f e .,V cos .' f ^ _ f gV,--. cos ay J 3 _ ^~ '" g'/ 2 --. cos ^> ^ ^ 



J 00 Ja + '/.jjr J 



and unless {(|-7r a) 2 cos a}" 1 is of the same or of higher order than (: cos)~'-, the 

 second and third integrals on the right are negligible in comparison with the first, 

 that is, if z n^ is of an order higher than z /3 , 



'/, -3 ^ = 2 V s (-12 cos a)' 1 '' = 2 1 V- (z cos )"'- e l/ * n , 

 and therefore, with the same restriction as to the magnitude of n, 



f" e i"+(+'/,)rf f ^ _ 2 V /S (z cos a)~ 1/2 e [--+(+) 1 V-(+V J ).] > 



the term of highest order only being retained. Hence 



VM = (cos a)~ 1/2 gt-*+ 1 Aw-(+Wi# 



* The parts contributed by the second and third integrals in the first expression for v-iu are of 

 order z~ l and therefore negligible, 



