194  Mr.  J.  W.  Nicholson  on  the  Diffraction  of 
Taking  the  centre  of  the  sphere  as  origin  of   polar  coor- 
dinates, and  putting  cos  0=/jl.  then  thfe  velocity  potential  in 
an  incident  plane  wave  travelling  along  x  may  be  expressed  as 
0=tf*to*-«o        ......     (1) 
Omitting  the  time  factor,  this  may  be  written* 
<£  =  A0-rA1P1(/a)  +  A2P2(/*)  +  ...  +  A„P„l»+...    .     (2) 
where  Ai=(SM-l);f(^Y.J»+i(ir).    ...     (3) 
The  difFractive  effect  is  derived  from  a  velocity  potential  -*|r, 
where 
or 
since  yfr  must,  like  <£,  be  a  function  of  zonal  character.     We 
deduce  at  once 
K+AW' (4) 
where 
g2(^n)  +  (^_'l^+1),(tn)  =  0.      .     .     (5) 
This  is  satisfied  by  the  Bessel  function  ^rn=r-*J«+^(*r), 
but,  for  the  present  purpose,  we  require  a  solution  finite  at 
infinity,  corresponding  to  a  very  large  value  of  k.  The 
ordinary  asymptotic  expansions  for  Bessel  functions  cannot 
be  used,  as  they  take  no  account  of  the  cases  in  which  n  is 
very  large. 
Make  the  substitution  f 
rfn=e±^,„ (6) 
where  (<w,  <j>n)  are  functions  of  r  and  n. 
Substituting  in  the  differential  equation,  and  equating 
terms  of  different  order  in  k  separately  to  zero, 
-fO-nS1)^  •  •  •  <8> 
where  a  term  <pn"  has  been  neglected  in  comparison  with  k. 
The  last  equation  gives 
fa>='(jfcV»--n  .  n+  l)*-(w  .  rc  +  1)4  cos"1  C^jfi^f-       (9) 
*  Lord  Rayleio-li,  foe.  cit. 
t  Webb,  Proc.  Roy.  Soc.  vol.  lxxiv.  p.  315  (1901). 
