ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 117 



z and z t . In this case the approximations to be used for </> and x<> are those given by 

 relations (x.) of the Appendix ; from these 



; 



1 t tan (<o+X<>) ' 

 that is 



+ &c., 



where T is a large negative quantity. Hence 



*-w = 2t sin <e-'*' + ie 2T e-' w ' + '''' ) + &c., 



that is 



gi-^. 



and 



f'2 4- 1 ^ "R' /2 "R V ' VWW + f.'f^o+SXo-f'-Wl = / 



I Z//(.-|- 1 I AV A\n iC T& / r l <> 1 O 1 i/ c l 



j smh 8 sum ! } /2 



NOW 



r l >r>a, 



therefore 



and - 



hence the order of the corresponding term in i//, compared with the principal part of 

 r/!, is (K-r)~ 1/2 , and is multiplied by an exponential with negative exponent. Further, 

 the portion of the series containing these terms is simply oscillatory on account of P n , 

 and therefore the sum of any number of these terms is not of higher order than 

 (KT) Vtyi by the appendix ; hence the part contributed by the terms of the series, for 

 which n + ^ Zj is of the same or higher order than Z/ 3 , is negligible compared with ^ t . 

 Again, the terms for which n + ^z is of the same or higher order than z'\ but 

 H + ^ZI is of lower order than Zi h , are at most of the order (/cr)" 1 ' 1 '^, and therefore, as 

 above, their sum is negligible in comparison with i//!. Similar results hold when 

 ?>?-!. Hence the part of \jj, if any, which is of the same order as T//J, is contributed 

 by terms for which n + is less than z or exceeds z by a quantity of lower order than 

 z 1 '' 3 when ?> rj, and by terms for which n + ^ is less than z l or exceeds z l by a quantity 

 of lower order than z/' 3 when r > r^. Different treatment is necessary according to 

 the form of approximate value of P n (/A) that is appropriate. 



3. When 6 is small, the approximate value of P n (/x) is given by 



whence to the same order 



SP 



= n + cos cosec 1 n+l sn 



and the series for \p is approximately 

 ^ = ~ 



K' 



when 



sn cos 7i + 1 e l -' + e' + ^--' 1 2n+ 1 sn 



