Potentials,  JEolotropic  and  Isotropic.  579 
Xs'  (#),  introduces  the  factor  m  in  the  first  step  where  (10  6) 
is  quoted,  with  an  identical  result. 
For  the  volume  potentials  of  magnetic  type 
jf"«:(i-«0(«+Vy+/8'*)-^ 
and 
f"(lr-0*(^  +  7^  +  ^)--^ 
which  we  may  call  U7(a?),   VS'(V)  tne  composite  terms  in 
energy  are  respectively 
3p2r0L0/(2s  +  2*'  4  3)  (25  +  2/  +  5)(2s  +  2/  +  7)     .   (19) 
and 
3p2t0L0       2.4...27+^__ 
8       "3.5...27+27T3 V     ; 
Tn  lieu  of  total  charge  in  connexion  with  ^,  we  may  here 
take  a  magnetic  moment*  defined  for  %s(.r)  by 
-J"- 
,0  fa-         'Vo        . 
u  d\ 
VJ" 
for  the  function  U  '  {x)  it  is  0   -77 ~  .      Thus    an    exact 
s  x  ■■  2s  +  3  .  2s  +  0 
correspondence  is  given  between  the  two  types  of  potential. 
§  6.  We  now  consider  with  reference  to  a  potential 
P  =  *1U0  +  *SU1+..:*nUn-l 
=    4]  (l-^a)(/,-l  +  ^a  +  ^+...^ 
the  problem  of  determining  the  k's  so  as  to  make  the  energy  E 
a  minimum  subject  to  the  constancy  of  total  charge  e,  E  and  e 
being  given  by 
^-""^  Lino ^ 37577  +  "57779  +  -J) 
L  .(21) 
-^.Ori+o +  •■'■]  3 
*  In  the  case  of  JE.  (117)  where  these  functions  are  used,  it  is,  however, 
a  question  of  total  charge,  and  e  =  3pr0/2  for  the  conductor  (s=0),  c=pT0  for 
the  uniform  volume  distribution  (s=l).  The  energies  for  Xo  an<i  f°r  Xi 
are  then  respectively  e2L0/36r0  and  <'2L0/70r0 ;  thus  if  we  seek  the  energy 
of  convection  currents  due  to  surface  charge  instead  of  body  charge,  we 
must  write  in  JE.  (117)  e2/36V2  outside  the  square  bracket.' 
