Short  Waves  by  a  Rigid  Sphere.  201 
Evaluation  of  the  Integrals. 
The  form  of  ^r  given  in  (24)  may  be  written 
-f=Ii  +  I2  +  I3  +  I4, (28) 
where 
*(Il5L)  =  e     4|   Ve+^2-x2-kc(l-x^]dx 
xexp.  ikc\  2x/l  —  x2  —  2%  cos-1  j;  +  -^ 
-A^rl-x*  +  xcos-l~±6x}.     .     (29) 
(I3,  I4)=^T(  dwVe  +  T  ^2-x2-kc{l-x2)^\ 
Jc  L  J 
exp.  ^c^cos-1^  -  \J~  -x2±6x  +  ~y     .     (30) 
The  quantity  U  denoting 
ITT 
/_'2kcd^\i e~*  1  x  " 
Uw;  •(1_^)i}2_.),+t,(,(1_^)rp-log-i+e-  <8i> 
Now  consider  the  integral 
1=  j  uetkcvdx (32) 
where  (w,  v)  are  functions  of  x  occurring  in  any  of  the  four 
integrals  above.  If  accents  denote  differentiations  with 
respect  to  x, 
ike  |  v1        j  c      ike  J  c  c/ct'  \  v' ) 
In  each  of  the  four  cases  above,  the  latter  integral  may  be 
shown  to  vanish  in  comparison  with  unity  *. 
Thus  -,    ,  > 
I=i<B<H  • (33) 
tkc  )  v 
*  Integrals  of  tliis  class  are  considered  in  the  paper  quoted  in  the 
previous  footnote. 
