Short  Waves  by  a  Rigid  Sphere.  197 
or,  i£  *61  be  the  argument  of  the  trigonometric   functions, 
the  numerator  is 
_i_  (2ii  +  X)t'1 
— n    3TTi —  { (H  +  iv2)  cos  6,  +  lH5  sin  01 } : 
and  therefore,  returning  to  the  old  notation, 
n  «.4.  n  ,»   «+*-  -f  (^'^-'' •"  +  !)«»  gi  +  «(^»-n.  n+l)l  si„  tf, 
l«.--(A«  +  l>«  •«  ^  2AV-«  .n+l  +  t(AV-n(n  +  l))i 
.     .     .     .     ( 
and  the  velocity  potential  of  the  diffracted  waves  is 
0 
or 
o(AV.#r»— n.n  +  l)i 
*=^TfP"(*L.      .     .     .     (16) 
at  the  point  defined  by  (V,  jju)  referred  to  the  centre  of  the 
sphere  as  origin,  and  the  direction  of  the  incident  waves  as 
initial  line. 
The  approximate  formulae  above  are  only  strictly  true  if 
kc  is  infinite,  but  are  very  accurate  if  kc  is  only  a  fairly  large 
quantity.  The  summation  with  respect  to  n  from  zero  to 
infinity  may  with  great  accuracy  be  replaced  by  a  summation 
from  zero  to  an  integer  close  to  the  quantity  kc.  Owing  to 
the  fact  that  kc  has  been  really  treated  as  an  infinite  quantity, 
all  the  values  of  n,  which  can  give  rise  to  an  appreciable 
effect  in  the  diffracted  wave,  will  be  included  in  the  new 
range  from  zero  to  kc.  Terms  near  n  =  kc  are  reserved  for 
future  treatment. 
Now  n(w+l)  =  (»+$)*—£< 
n  +  -1- 
Denoting     ,    2  by  x7  we  note  that  «  is  a  quantity  rising 
by  increments  y-   from  ^-   to  1,  or,  to  a  sufficient  order  of 
approximation,  from  zero  to  unity. 
Accordingly  we  write 
(w,w  +  l)* 
~  IV  kc  J        4AVJ 
to  a  sufficient  order. 
kc 
kc 
