£76  Mr.  R.  Hargreaves  on  some  Ellipsoidal 
and  the  surface  integral  is 
i' 
*■'■'*  =  i0iy  ■_■  ■  ^" 
Now  the  potential  fa  has  a  surface-density  pur/2,  and  a 
volume-density  —s'pii~\  The  term  in  energy  due  to  the 
interaction  of  fa  and  fa.  is  therefore 
E(«,  0  =1  jpflf^S-  jVp<-2  f  sdr 
8     V2^  +  l      2s  +  127+27+T'      8(25  +  2V  +  l)    l     ; 
The  energy  due  to  ^s  alone  is  3p2T0</>0/16  (4s +  1),  £.*.  we 
write  s'=s  and  halve  the  result.     The  energy  belonging  to 
h^+Kfa  is 
16      Us  +  1      2*4- 2*' +  1   '  4/+1-J' 
but  it  is  clearly  sufficient  to  give  as  in  (14)  a  single  compo- 
site term.     The  total  charge  attaching  to  fa  is 
For  normal  distribution  on  a  conductor  s  =  0,  e  =  3pr0/2,  and 
the  energy  is  <?2<£0/12t0. 
From  (14)  may  be  derived  the  composite  term  in  the 
energy  of  ksTJs  +  ks,Usr,  or  of  k,(fa— fa+1)  +  ks,(faf— fa'+i). 
For  this  term  we  are  concerned  with  the  products 
Ps'  ^s—Ps'+i  tys-ps'  ^s+i+ps'+i  fa+u 
and  the  application  of  (14)  yields 
3pV0<£o  r  _     1 2  1        1 
^8~~  L2s  +  2s'  +  l      2* +  2*' +.3  +  2*-t-2*'  +  5j 
-  ^— -    3^-^ (15) 
2*  +  2s/  +  l  2s  +  2J+'d  25  +  26-'  +  5'     l     ; 
which  multiplied  by  h  ks,  is  the  composite  term  required. 
p  f  dX  '     1 
To  deal  with  the  function  Vs  or  j  |  — t=(1  —  ujs,  the  twTo 
summations    for    the    indices    of   p    and    i/r    may  be   taken 
