ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 141 



The value of x is given by the relations 



tan( + x) = 2e 4 "' / % or tan x = e 4 "*' J (ix.), 



the leading term only being retained. 



When n+^z is great compared with 2'% the necessary approximations are obtained 

 from the original formula for viu, viz. : 



v-iu = 2-'%-V' f-i fY u '""* +<ll+1 ' yfc rf0 + f e -- -"'*+<+'-*>* f fy 

 L J o Jo 



r* -| 



- 1 sin (n + ) TT\ e-~ ^ * cfy I . 



The principal part of this expression arises from the second integral, and is 

 contributed by the values of \ft near to the value that makes sinh />( + i) / 

 stationary ; if this value of ty is 8, then 



z cosh 8 n + ^, 

 and writing i/> = S + i// 1; it follows that 



-CO -00 



I --.rsinh^ + fB + Va)^ /./, I g(n + Vi) $-- ""h S- 2 - si h * sinn2 V 2 ^]H'' + ^) ( si 'ih *i~ W j^,/, 

 Jo J-S 



the principal part of which is equal to 



e (+'-^iniu f g-v^sinh w ^ _ 2' V >S 2~ V (sinh 8)-v 2e <+',y-smi.S ) 



J - 00 



f- 

 the part which depends on the integral being of lower order when n-\-^ z is of 



higher order than z' 3 ; hence 



v - 

 where 8 is given by z cosh 8 = 



To obtain u it is necessary to calculate the leading terms of the imaginary part of 

 the expression for viu, and this arises from the first integral ; the part required is 

 the principal part of 



-%i e- l -~ shl<>+(H+ ^' e dd-ii \ e'-"-<''+ 1 / s > l d0 - 



Jo Jo 



The exponent in the first of these is stationary when 9 = 18, and the exponent in the 

 second when 6 = tS, writing in the first 6 = iS 0j and in the second 6 = 18 + ^, the 

 expression becomes 



f fir- 1* fir + .S 



_ l. l g-(n-(-Vs)*+.rBhih4 I I g-2.-sinhsin 31 / 2 *i + i-cosh{(a l -sin 1 ) JQ .1 g -2- sinh sin 2 VA-'-cosh J (, sin 



U-d l J,s 



