198  Mr.  J.  W.  Nicholson  on  the  Diffraction  of 
^°  in  =  exp.  (ikcx  -  +  -j  \ 
co  =  (r2 -c~ 'x2 )\-cx  cos"1  — 
ft,c  =  c(l—  a2)?  —  CX  COS"1.?', 
((2  —  x2)cos0l  +  ikc{l  —  x2)%sin6l-^ 
'n  =  -  2Icc'r  \  ~  2-xr  +  ikc(\-x^  ~  ) 
X^T.  exp.  ike  \  ( I  -  «r>^  +  #  sin  ~ ]  #  }     •      (1?) 
where  ^       7    f  ...        9N1  ,    ")       it  n  QN 
^!  =  ^c^  (1  —  #2)W#cos -'a?  ^  —  2. 5     •     •      V1") 
Now  the  sum  o£  a  finite  series  2S(#)  o£  the  type  (16)  may 
be  shown  to  be  given  by  * 
2S(ff)-H2R^^-fs0c).L<fc, 
and 
where  /cc  is  large,  and  the  large  integer  less  than  he  is  r,  where 
r  =  kc(l—e),  in  which  e  is  small.  L  denotes  the  principal 
value  of  i        /  •       -,  ,    n 
R  is  a  residue  of  S(#)  at  any  pole  which  it  may  have  in  the 
contour  C,  a  unit  circle  round  the  origin. 
But  the  poles  of  S(a?)  in  this  case  are  all  complex,  being 
the  roots  of  the  equation  2— x2  +  ikc(l  —  ^2)t  =  0.  The 
corresponding  residues  cannot  contain  a  factor  of  the  form 
exp.  (kc<j>),  where  </>  is  real  and  positive,  from  the  form  of 
the  integral  and  from  physical  considerations.  The  imaginary 
portion  of  the  root  corresponding  to  a  pole  will  therefore 
introduce  a  real  exponential  factor  of  large  negative  argument 
into  E,  which  may  accordingly  be  neglected. 
Thus  the  velocity  potential  of  the  secondary  disturbance  is 
he2    ' 
r=-2-* 
7T  r    _ 
r  I  c^x      S  (^  —A'2)  cos  #!  +  iked—  x2)i  sin  Oi  )  p     (  x 
exp.^{(l-^-^cos-^+^-(^-^  +«cos-^  j,  .     (19) 
*  This  theorem  was  given  hy  the  author  in  a  paper  read  "before  the 
London  Mathematical  Society  on  Nov.  9th.  It  is  proved  by  the 
application  of  contour  integration  to  the  function 
:c  kc  Ac- 
/  X—j-        x~-~ 
y  kc  kc 
