J 
584  Mr.  P.  Hargreaves  on  some  Ellipsoidal 
The  residue  can  be  expressed  as  a  linear  function  of  the  P's 
of  odd  degree  from  P2w+3  to  p2»+"2p+l>  anc* tne  multiplication 
of  these  by  [jl-ci+1  and  integration  from  0  to  1  yields  a  zero 
result  from  q  =  Q  to  q  —  n.  The  integral  on  the  left  hand  of 
(30)  is  then 
—  (2q  -h  1)  —  (p  —  1)  (2ra  +  2/>  +  1)  =  rc(2rc  -f  2p  + 1)  —  2^  as  required . 
A  comparison  of  (29b)  with  (24)  shows  that  the  solution 
gives  directly  the  density.  Now  a  zonal  harmonic  has  a  point 
of  inflexion  between  each  maximum  and  minim  am,  therefore 
/*~1Pl2n+i  vanishes  for  n  —  1  values  of  p?  ;  hence  the  density 
vanishes  for  these  values  of  ua.  The  distribution  is  one  in 
which  there  are  n  concentric  ellipsoidal  shells  of  alternate 
positive  and  negative  charges.  When  n  is  indefinitely 
increased  the  charges  on  all  but  those  for  which  ua  is  nearly 
=  1  are  indefinitely  small,  and  the  case  becomes  that  of  a 
conductor.  A  simple  independent  proof  for  the  case  of  n 
infinite  shows  the  character  of  the  limit.     Since 
ap  +  ap+i+  . . .ad  inf.  =  -£ 
the  equation  of  condition  may  be  written 
i=£i(«P  +  ap+i  +  .'..)+fs(«p+i  +  Zp+2  +•••)  +  ...=171+^2+... 
i.  e.  it  can  be  expressed  in  terms  of  the  rfs.     But 
BD.  =  f lS7]1  +  . . .     subject  to      0  =  Br]1  +  Srj2  + ... 
makes  the  f 's  all  equal  ;  and  it  is  then  easy  to  show  that  each 
=.2(2p  —  1),  and  that  co=(2'p  — 1)/2.     The  potential  P  (case 
p=l)  is  then 
e 
d\ 
J  A  V  J  °ToJ, 
and  the  potential  Q  (case  p  =  2)  is 
^j0O(^  +  7V+^)(l-^)(l+Wa  +  ^+...)^ 
The  higher  derived  functions  which  would  correspond  to 
higher  values  of  p  in  the  minimum  problem  are  not  here 
constructed. 
§  7.  In  conclusion  we  return  to  the  general  theorem  of 
moments,  give  an  independent  proof,  and  then  show  how  to 
