Short  Waves  by  a  Rigid  Sphere.  203 
whence  we  finally  obtain 
n  exp.ikc\7r  —  0—  sin-1—  \/%  —  1 1 
_  /  c  \  i  6  4  I  ■  \  r         V     r-  / 
*  ~  ~~  IJttAt2  sin  £  J   '  /        c2^ " 
7r  —0—  sin-1 
r 
exp 
+ 
xp.  —  ikcl  —TT—e  +  mi-1-^  +\/~2  — l) 
i  e 
+  - 
xp.  tkcl  7r  +  0  —  sin-1-  —  a/  -^  —  1 ) 
7r  -f  0  —  sin-1- 
i  exp.  —  ikc(6—TT-\-  sin-1    +  \/^  —  1) 
+ v -   v  c    -/  > 
7r  -+-6—  sin-1- 
which   holds    except   close    to    the   axis    of   w,    where    the 
asymptotic  expansion  of  the  spherical  harmonics  is  not  valid. 
At  a  great  distance, 
,fr_  _  /         g         \*  * <(4  -*0  J  exp.  tfecQ—  6)       e^p^Mnr  +  d)  \    m) 
V-      Wkr^me)  I  7T-0  TT  +  0  i   v 
The  more  general  formula  reduces  to 
c        ^         ^  fexp.^^Tr  +  ^-sin-^-Y  '-,-l) 
exp.^-0-sin-^-^  -l)| 
7T  — 6/  — sin-1-  ) 
For  points  on  the  positive  side  o£  the  axis  of  a?, 
whereas  the  asymptotic  expansion  gives 
/        2        Ycostt 
\&C7r#sin0/       ^4  " 
