1^03.] Electric Waves round a Conducting Obstacle. 65 



therefore 2-^- is negative and at most of the order (>i + *)-J, 

 T,;(^) being at most of the order unity; further, when ; = ;i + i, 

 r- -ilog3, and as n increases T, t diminishes rapidly, hence' 

 cosh 2r /t cos- xn and sinh2r H sin-' x/l are less than unity and diminish 

 rapjdly as n increases ; also writing n + J = z cosh 8, 



r n = - -J log 2 - zS cosh S + .: sin 5, 

 3r n 

 aT ! ~ 8 > 



hence, unless 6 is small, r n is a large negative quantity, and therefore 

 cosh2r n cos'x, sinh2T cos'xn, by the above, very small, whence it 

 follows that c x /a is always very small. The series So an d S fl there- 

 fore both vanish. 



It is known that neither u n (z) or v n (x) can vanish for a value of w , 

 which satisfies the condition that n + J > ?5, and the Iagt fcime ^j 



vamshes as H increases the value of ^ isL ; Jtt n increases farth 

 *. diminishes to zero. When *-(* + i) is of hi hcr onj fa f 

 writing N + J = s sin a, where 'TT > a > 0, 



a - 



2/fc ' 3/r- c cos a ' 



hence ^ is negative, and as increases - ^ diminishes and tend, 



Oil 



to the o,xler ( f .)-i When , - ( + 4) is of lowor order tfaan ^ 

 ^ is given b>- 



whence, remembering that R' w is negative and R a is of the order 

 {H + ^' "^ iS n ^ ative and of the order (n + i)-.?. When w + $ > :, 

 using the relation tan < = r"^ f 



and therefore 



3^7i = 



a/?/ 



where 



.-cosh 8 



