ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 139 



becomes 



<) = z sn 



now 



zn\ = z(l cos e) = 



therefore 



and 



R = z {z 2 -(rT+l) a }-' /2 = z {z-n-i}-'' {2z-(z-n-$)}- l/ > ; 



that is, retaining the principal part only, 



The leading term in (v M) z~ l/e being known, a further approximation can be 

 obtained from the differential equation 



the result is 



,1.5 i 1.5.7.11 



The same remark applies to the approximation for v iu when zn^ is of higher 

 order than z ', the ordinary differential equation for BESSEL'S functions being made 

 use of. 



When n + ^z is of the same order as 2 1/3 the corresponding series when /A is positive 



c 



is required. The principal part of v iu now arises from the integral e tVf " f:< c/, the 



Jo 



principal part of which is contributed by values in the neighbourhood of the value of 

 that makes 3/i 3 stationary; that is, in the neighbourhood of = //-', and 

 writing 



= /* l/a +i, 



it follows that 



f <P*-P d = e 2 " 3 ' 3 f e-^' 1 -* 1 d{,, 

 Jo J-it 1 '" 



therefore the principal part is 



This result gives the leading term in the value of v, to obtain the leading term in 

 the value of u it is necessary to calculate the principal part of the imaginary terms, 

 that is, the principal part of 



Of. STOKES, 'Camb. Phil. Trans.,' vol. x., p. 105. 

 T 2 



