fine Lines or transmitted by narrow Slits. 351 



The secondary waves diverging from the obstacle are 

 represented by 



^ = B D (A'r)+B 1 D 1 (^)cQS^ + B 2 D 2 (*r)cos2^+ ..., (3) 

 where 



^)=-(a i -M 1 -r7fc+i^W----} 



= (7+ log J ){l- J + %£#- ■ • •} 



7 being Euler's constant (-5772 . . .), and the other D's are 

 related to D according to 



D„(--) = (-2--)»(^)" D » W (5) 



The first expression in (3) is available when z is large and 

 the second when z is small. It should be remarked that the 

 notation is not quite the same as in the papers referred to. 

 The leading term in D when z is large is 



D ^=-ffi v "' •••••• < 6 > 



and in finding the leading term in D n (z) by (5) it suffices to 

 differentiate only the factor e~ iz . Thus when z is great 



DJz] 



-*• (£)**-* co 



Accordingly when in (3) yfr is required at distances from the 

 cylinder very great in comparison with \, we may take 



+=-(^y«- ar {B +t.B 1 oos^-BjCos2^+...}. . (8) 



We have now to consider the boundary conditions to be 

 satisfied at the surface of the cylinder r = c, and we will take 

 first the condition that 



<jM-^=0 (9) 



at this surface. We have at once from (1) and (3) in virtue 

 of Fourier's theorem 



B^— 2tJ 1 (Ae)-r-D 1 (*c), 



and generally 



Bn=-2i n Jn{ke)+T)n(kc) (10) 



2B2 



