Potentials,  jEolotropic  and  Isotropic.  581 
The  conditions  o£  no  variation  are 
7Jlz=cob1,   n2  —  cob2, . . .with  l==bp^i+ . .  .bp+n-\^n  ; 
i.  e.  apgi  +  . . .  Op+n-i  &  =  «^p 
>•     •     ■'   •  (25) 
«j9+7i-  i  f  i  +  •  •  .«p+2»-2&»= cobp+n-i 
bp%i+...  bp+n-l    ?7i=l 
Write  the  first  as 
and  note  that 
a^  +  a^+i  +  . . .  &p+»-i  =  i  [bp— bP+n) 
_  i  r        i        _  i  I 
4  \  2p-l  2p  +  l      2p  +  2n  — 1  2p  +  2n  +  lf 
_  i  <r   i      _i i_  i      •) 
"  8   l2p-l      2/7  +  1      2p  +  2n-l  +2p-f27i  +  l  j 
*  f  1  1        =1 
4  \2p-l  2p  +  2™-l       2p+l  2p  +  2rc  +  li  ' 
The  first  equation  of  (25)  is  then 
i    r   «i    ,     ^    ,  #»      I  _     i    r  &-, 
2p-lL2p.+  l   '    2p+'3  2p  +  2n— lj       2p  +  lL2p  +  3 
+  -2i,  +  2n  +  i]  =4^=2ft,(^Tl  -  2^Ti) 
and  others  o£  this  type  are  got  by  increasing p  by  1,  2,  3... 
Since  &;>  +  bp+  \  +  . .  .bp+n-i  =  — ■  ,  the  equa- 
2p  —  1 .  2/?  +  2m  —  1 
tion  o£  condition  treated  in  this  way  becomes 
+  -^T^  +  ---o„  ,  o„     i  =^-1.     .     (2b) 
2p  +  l       2p  +  3  2p  +  2n  —  1 
The  group  of  equations  is  then  equivalent  to 
2^i+-2^;i^=2"+c^-1)' 
