568  Mr.  R.  Hargreaves  on  some  Ellipsoidal 
(2)  Different  equations  o£  state,  all  of:  which  are  fair 
approximations  to  the  behaviour  of  a  real  gas,  indicate 
very  different  values  for  these  inversion  temperatures. 
(3)  The  sensitiveness  of  the  positions  of  these  points  to 
change  in  the  characteristic  equation  of  the  fluid  makes 
a  knowledge  of  their  actual  position,  as  determined 
experimentally,  a  very  valuable  means  of  discriminating- 
bet  ween  the  relative  validity  of  any  proposed  equations 
of  state. 
In  conclusion  attention  may  be  drawn  to  the  fact  that  all 
the  results  given  in  this  paper  are  exact  consequences  of  the 
equations  of  state  to  which  they  relate. 
XLVIII.  Some  Ellipsoidal  Potentials,  JEolotvopic  and 
Isotropic.     By  B.  Hargreaves,  M.A* 
IN  a  former  paper  on  iEolotropic  Potential,  the  potential 
functions    corresponding  to   normal   distribution    on    a 
conducting    ellipsoid    and     to    uniform    volume-distribution 
(l-ya)d\ 
VJ       ' 
where  J  is  the  cubic  in  A  which  replaces  the  isotropic  form 
(a2-\-X)(b2  +  \)(c2-\-\),  and  va^=l  represents  the  seolian 
quadric  which  replaces  a  confocal.     If  the  additional  factor 
..  a  occurs  under  the  integral  sign  a  vector  type  is  given, 
QX 
which  was  also  considered  in  the  second,  but  not  in  the  first 
were  treated.     The  functions  are  I      — — -  and.   I 
Ja    vJ       J 
case. 
It  is  proposed  now  to  deal  with  the  more  general  types 
j: 
"/jK>     or     «a(l-Ma),     or     (l-wa)s], 
and  with  types  in  which  the  factor  -^— ^-  is  attached  to  the 
QX 
several  forms.  The  second  and  third  forms  belong  to  volume 
distributions  with  densities  depending  on  powers  of  ua;  the 
first  has  also  a  normal  surface-density. 
The  energies  attaching  to  these  potentials  take  the  simple 
form  of  numerical  multiples  of  that  of  a  conductor.  They 
are  obtained  by  the  use  of  what  (in  the  paper  cited)  were 
called  wave-forms  of  the  potential.  Two  theorems  are  re- 
quired for  the  purpose  :  one  to  connect  the  wave-forms  with 
*  Communicated  by  the  Author. 
