Short  Waves  by  a  Rigid  Sphere,  195 
The  general  solution  of  (5)  is  of  the  form 
=       A^  +  Bg- 
'    >l  I  7.9  .9        7  9 
We  now  proceed  to    determine   the  values  of   A  and  B 
making  this  identical  with 
Jn+±(kr) 
(feO* 
By  Lommel's  theorem,  when  z  is  large,  we  have 
T/x      /2\*-      /       77         tt\     r-      l2-4rc2.32-4rc2  ,         | 
/2\*.   /       7T        tt\    rl2-4rc2      I2_4rc2.32-4rc2.52-4rc2  \nm 
whence,    retaining    only    lowest    powers    of  -,  our   solution 
must  be  identical  with 
fffi-aHfr--"-?)-^-1-'^'"*'} 
Since 
(£V  -n  .  n  +  l)i       (jfrr)  *  I  X  +     4#V    +  *  '  "  I 
Therefore 
whence  on  substitution 
and  finally,  when  /bj  is  large,  and  no  restriction  is  laid  upon 
the  value  of  n,  except  that  it  be  less  than  kr, 
f    /7  9    9  r1\I  ,  ,.v  ,   (ft.  7I+  I)'"' 
JU(fe-)  =  /2 \*  Q08{(fr*—«.n+ !)■-(!.  + A)  cos-i^^- 
(*)•)*         W"  {(A»r»)(AV— ».n  +  l)}i 
0  2 
(12) 
