580  Mr,  R.  Hargreaves  on  some  Ellipsoidal 
If  a  similar  problem  is  stated  for 
Q 
dX 
Hj  &  +  WK.  /.Vv:»-1)(^  +  7'i/  +  /3'4  -^, 
<£0  is  replaced  by  L0  and  e  by  mx,  and  the  coefficients  open  with 
1  ,     1      .     .      ,     „        1  ,1 
«  and  rr 
instead  of 
and 
^  = — -•      By  writing 
3.0.7  3.5  1.3.0  1.3         J  ° 
A-=  t-^—  for  each  subscript  the   minimum    problem  is  pre- 
'6PTo  .  . 
sented  with  pure  numbers,  and    we  may  generalize    it   by 
making  the   series  of  coefficients  start  at  a  different  place. 
Thus  nsing 
^=l/(2p-l)(2p+"l)(2p  +  3)    and  bp=l/(2p-l)  (2p  +  l), 
it  is  required  to  find  w  the  minimum  value  of 
ft  =  f !?!  +  &%+..,  Ert. 
where  «7i  =  «?&  +  «p+i&  +  ■  •  •     Op+»-i?« 
(22) 
define  the  ?/s,  and  1  =  ^f ,  +  &p+i?2  +  •  •  •    &p+»-i?« 
is  an  equation  of  condition.  From  the  point  of  view  of 
potential  theory,  interest  attaches  to  the  value  of  the  minimum, 
which  proves  to  be 
2p  —  l    n  +  1  '2n+2p  —  l 
•> 
(23) 
n  2n  +  2p-\-l 
in  the  general  case  ;  and  to  the  nature  of  the  distribution 
which  gives  the  minimum.     For^=l  the  case  of  (21), 
w=(n  +  l){2n  +  l)/2n(2n+3)  and  E  =  ^0o)/6r0  ; 
while  the  density  for  the  potential  P  is 
W\J i  -.6+8  (ft  -  ft)  «. + 3(6-  6)  «;+■•  +  »6< 
,-nl 
or  say 
>   (24) 
