Short  Waves  by  a  Rigid  Sphere.  205 
dlTT 
[U]c=-_^_ '.     (44) 
Thus 
i.=-[u]c. 
.  —  J-cl  —  2tt  +  \/  -t2—  1+  sin-V-j 
ike 
exp 
\  »     I.- 
c 
2tt+  sin-1 
exp.  —  tfo(  —  2-7T+  sin-1-  +  A /™2  —  1  ) 
c i_ 
(  1 5  2tt—  sm   L~ 
\        r2  J  v 
Similarly, 
c  x  exp.^(27r-sm-^;-^/^-l^ 
|1 s)  27T—  sm   1 
Thus  on  the  negative  portion  of  the  axis  of  a, 
exp.  ikcl  27T  —  sin-1 
.  c    1  I  V r 
27T  — sm   x- 
exp.  ike  [  27r  —  sin- 1  ~  —  \  /  ~2—l 
2tt—  sin-1- 
r 
and  at  a  great  distance  along  this  axis, 
G    r 
1    ^2mkc  _l_  ,#2irikc 
J 
(45) 
^=   __   J    e2?nke+LeMc   I 
=  — ?=exp.2**CTr.lJ lZ.    .     .     .     (46) 
7TY/  L  v 
The  values  of   i|r   here  given,  with   a  time   factor  e-**v* 
added,  correspond  in  each  case  to  an  incident  disturbance 
<£=  exp.  ik{x  —  Yt). 
The  greatest  diffractive  effect  occurs  in  the  directions, 
on  either  side,  which  lie  in  the  line  of  the  incident  disturbance, 
and  the  effect  near  the  axis  of  the  disturbance  is  of  a  higher 
order  than  that  at  other  inclinations.  The  formulae  are  not 
true  close  to  the  sphere,  as  the  asymptotic  expansions  are 
not  valid  there.  They  give  the  effect  of  all  harmonic  terms 
not  close  to  n  =  kc. 
