Potentials,  ALolotropic  and  Isotropic.  569 
those  written  above,  and  the  other  to  deal  with  moments 
and  products  of  inertia  of  any  even  order  for  an  ellipsoid. 
The  evaluation  of  energy  turns  on  a  similarity  in  the  forms 
of  the  two  theorems,  and  they  prove  to  be  closely  related. 
A.  potential  is  then  considered  which  is  a  linear  function 
of  members  of  the  second  type,  namely, 
i: 
0  7-y 
(l-UaXk1±k2Ua+kBU2a+    .  .   knUna-1)— J- 
The  energy  is  a  quadric  in  the  Fs,  persymmetric  in  form  ; 
while  the  whole  charge  is  a  linear  function  of  the  £'s.     For 
any  value  of  n  this  potential  may  be  determined  so  as  to  give 
a  minimum   of  energy  subject  to  constancy  of  total  charge. 
The  minima  have  simple  values  which  decrease  as  n  increases, 
ranging  from  that  for  uniform  volume-distribution  (n  =  l)  to 
that  for  normal  distribution  on  a  conductor  (?>  =  co).     Thus 
the  potentials  form  an  interesting  series  of  links  connecting 
the  two  potentials  commonly  considered  with  reference  to  an 
ellipsoid  ;  and  each  potential,  with  a  distribution  depending 
on  n  constants,  in  some  measure  simulates  that  of  a  structure 
containing    n    parts   brought  into   relation    by    a   minimum 
condition. 
To   give  greater  generality  the  work  is  written  in  seolo- 
tropic  form  ;  but   most    of  the  energies   evaluated  vary  as 
°°   d\ 
— j-=r,  and    the    only    difference    between  isotropic  and 
geolotropic  forms  consists  in  the  relation  of  the  constants  in 
the  cubic  J  to  the  axes  of  the  ellipsoid. 
S  1.  Of  the  three  forms* 
j 
(too         S    7\  Z^00         S 
(1  —  ua)dX 
(l—ua)sd\ 
(1) 
the  first  will  be  taken  as  fundamental,  results  for  Us  and  Y s 
being  directly  deducible  from  those  for  \jrs.  Since  ua  may 
be  written  =1  when  it  is  not  under  the  sign  of  integration, 
we  have 
dtys  _  p      dX     sp  r°°   s_1  "dua   d\ 
dx  ~~  ~  ~ivl  d®  +  4  JA   Ua      §*  ^/J ' 
*  The  notation  is  that  of  the  previous  paper  on  '  yEolotropic  Potential ' 
in  Phil.  Mag.  April  1905,  quoted  as  JE. 
Phil.  Mag.  S.  6.  Vol.  11.  No.  64.  April  1906.         2  P 
