200  Mr.  J.  W.  Nicholson  on  the  Dif Taction  of 
and  if  r  is  great  compared  with  dimensions  of  obstacle 
♦"-OS.)'-"^!""-— (--9 
xexp.  Lkc{(l-jr)^-d-cos-\v  +  d"7r}cli\    .      .      (25) 
where  P(#)  denotes  the  function 
(2-.r2)  cos  g!  +  ife(l— jp8)*  sin  0X  .„ 
2—^  +  ^(1-^)1             J     *     *     l     j 
which  always  lies  between  positive  and  negative  unity. 
To  obtain  an  integral  formula,  free  from  the  zonal  harmonic 
term,  and  suitable  for  the  case  of  points  near  the  axis  of  a, 
we  note  that  the  asymptotic  expansions  of  P;z(cos  6)  and  of 
J0(~)  are  respectively 
\/^0-oos^-e-i)' and  \Zh"*('-i)' 
and  therefore  when  n  is  great 
Pn(cos^)=Jo(27ism|) (27) 
We  therefore,  under  the  integral  sign  in  (19),  merely  replace 
P         (/*)     by     J0f2tesinf) 
to  obtain  a  simpler  form. 
On  the  harmonic  terms  for  which  n  is  greater  than  kc. 
The  quantity  a>c  now  becomes  imaginary,  and  the  expo- 
nentials of  the  previous  investigation  become  those  of  real 
quantities.  Since  i/r  cannot  be  infinite  at  great  distance,  the 
exponential  of  positive  argument  cannot  be  present.  The 
remaining  one  of  order  e~kc  appears  as  a  multiplier  in  yjr,  and 
the  portion  of  yjr  due  to  terms  for  which  n  is  greater  than  kc, 
is  therefore  vanishingly  small  in  comparison  with  the  portion 
previously  found.  This  supplies  the  justification  for  taking 
the  limits  of  x  as  zero  and  unity  in  the  summation.  The 
terms  near  n  =  Jcc  are  not  accounted. 
