Short  Waves  hy  a  Rigid  Sphere.  199 
where  T         1   ,         ,,        .       x 
This  formula  is  of  most  interest  in  the  case  of  points  lying 
along  the  axis  of  x,  for  the  Legendre  function  then  takes  the 
values  (±l)?i. 
For  a  point  at  a  great  distance  r  on  the  side  from  which 
the  incident  waves  come;  expanding  the  terms  containing  r, 
i7r/e-Jer\r  f(2-x2)cosdl  +  ikc(l-x2flsm01) 
xexp.  ikc{(l— x2)i  —  xcos-^  +  xtt],      .     .     (20) 
On  the  opposite  side,   P         (/*)=("")" 
3i7r  — (§-J-£c#; 
and  +=-  2^.,-t  (— sr-JJ cL*rfn 2-^  +  «fe(l-^ 1 
x  exp.  iA;c-|  (1— x>)\  —  xco^lx  +  2xir\  .     .     (21) 
Round  the  circumference  of  the  sphere  the  integral  takes 
a  much  simpler  form,  but  is  invalid.  >i 
Returning  to  the  formula  (19),  we  note  that,  using  the 
polar  angle  c£  as  variable  (x  =  e^)}  the  contour  is  traced  by 
making  cj>  range  from  zero  to  2tt,  x  takes  the  value  unity  at 
both  limits  (whilst  its  logarithm  has  the  values  zero  and  2ttl 
respectively).  But  the  value  x=l  corresponds  to  a  large 
value  of  the  integer  n  =  —  J  +  hex. 
We  therefore,  with  great  accuracy,  may  write  the  asym- 
ptotic value 
p-^K™)*°"(("+fi'"0-  •  (22) 
throughout  the  range. 
In  terms  of  x, 
P         (,*)=( l^\%0Jkcx6-y\     .       (23) 
Therefore  at  a  point  (r,  6)  not  close  to  the  axis  of  x 
^=     /  8/^8    yjrf     Lafcfo       r  (2-,^cos(91  +  ^Kl-^)"sin^. 
\irsin0)  r  Jj^cW I  2-s*  +  dw(l-s»)* 
Xcosffa?0—|j.  exp.tfo  j  (1  —  .r)^  —  wcos"1.;'  4-  Tjj'  — (  V~''"j 
+  .rens-1-  I,     .     .     (24) 
