Potentials,  jEolotropic  and  Isotropic.  575 
Thus  the  three  forms  all  depend  on  one  angular  integral 
which,  in  view  of  the  value  of  R,  and  t0  =  473-/3^  Aff,  is 
1    C(  l\  +  m/Jb  +  nv)2s  dco       -j  uA  (I,  m,  n)  j-® 
This  integral  is  a  variant  of  (4  6)  got  by  writing  a  for  A, 
A  modification  of   (11)  by  a  factor  ax  +  c'y-j-b'z  is  required 
for  the  %  functions.     Operate  with  a  -j-f  +  cf  -j f-  b'  -j-  on  (11), 
cti  elm  an 
and  raise  s  by  1,  thus 
-  i  {ax  +  c'y  +  6'~)  (&c  +  m#  +  ws) ps+1  dr 
To  J 
(2i  +  3)(2*  +  5)UJ  '    '     '     l     ^ 
-  I  (ax  +  cfy  +  b'z)(lx  +  my-\-nz)2s+1uSadT 
=  (2s  +  3)(2s  +  2s'  +  5)  (4)  '  (m) 
i  r 
-  I  (a^  +  c/?/  +  ^)(^  +  my  +  ^)2s+1^iS 
-iSi© (12-> 
are   connected   with   (12)   in  the  manner   shown   above  for 
(11  b)  and  (lie). 
§  4.  The  evaluation  of  energy  now  follows  by  combining 
the  typical  integrals,  thus  : — 
=  (2s +  1)  (2s +  2*' +  3)  J  29rwpwAV3by(m) 
and 
3*o 
(25  +  1)  (25+ 2s' +  3) 
using  <£0  as  in  iE.  (135)  for 
(13) 
Jo    \/JJo    -v/S^O  ~J  2 
We  may  also  write  (13)  as 
7rupuA]'"2 
J 
""  *•  ''T  -  4(2s  +  l)(2s+ 27+  3) : 
(136) 
