﻿Aepinus 
  Atomized. 
  267 
  

  

  extreme 
  interest, 
  geometrical, 
  dynamical, 
  and 
  physical, 
  but 
  

   cannot 
  be 
  pursued 
  further 
  at 
  present. 
  

  

  § 
  17. 
  To 
  guide 
  our 
  ideas 
  respecting 
  the 
  stable 
  equilibrium 
  

   of 
  moderate 
  numbers 
  of 
  electrions 
  within 
  an 
  atom, 
  remark 
  

   first 
  that 
  for 
  any 
  number 
  of 
  electrions 
  there 
  may 
  be 
  equili- 
  

   brium 
  with 
  all 
  the 
  electrions 
  on 
  one 
  spherical 
  surface 
  concentric 
  

   with 
  the 
  atom. 
  To 
  prove 
  this, 
  discard 
  for 
  a 
  moment 
  the 
  atom 
  

   and 
  imagine 
  the 
  electrions, 
  whatever 
  their 
  number, 
  to 
  be 
  

   attached 
  to 
  ends 
  of 
  equal 
  inextensible 
  strings 
  of 
  which 
  the 
  

   other 
  ends 
  are 
  fixed 
  to 
  one 
  point 
  C. 
  Every 
  string 
  will 
  be 
  

   stretched 
  in 
  virtue 
  of 
  the 
  mutual 
  repulsions 
  of 
  the 
  electrions 
  ; 
  

   and 
  there 
  will 
  be 
  a 
  configuration 
  or 
  configurations 
  of 
  equili- 
  

   brium 
  with 
  the 
  electrions 
  on 
  a 
  spherical 
  surface. 
  Whatever 
  

   their 
  number 
  there 
  is 
  essentially 
  at 
  least 
  one 
  configuration 
  of 
  

   stable 
  equilibrium. 
  Remark 
  also 
  that 
  there 
  is 
  always 
  a 
  con- 
  

   figuration 
  of 
  equilibrium 
  in 
  which 
  all 
  the 
  strings 
  are 
  in 
  one 
  

   plane, 
  and 
  the 
  electrions 
  are 
  equally 
  spaced 
  round 
  one 
  great 
  

   circle 
  of 
  the 
  sphere. 
  This 
  is 
  the 
  sole 
  configuration 
  for 
  two 
  

   electrions 
  or 
  for 
  three 
  electrions 
  ; 
  but 
  for 
  any 
  number 
  ex- 
  

   ceeding 
  three 
  it 
  is 
  easily 
  proved 
  to 
  be 
  unstable, 
  and 
  is 
  there- 
  

   fore 
  not 
  the 
  sole 
  configuration 
  of 
  equilibrium. 
  For 
  four 
  

   electrions 
  it 
  is 
  easily 
  seen 
  that, 
  besides 
  the 
  unstable 
  equilibrium 
  

   in 
  one 
  plane, 
  there 
  is 
  only 
  the 
  stable 
  configuration, 
  and 
  in 
  

   this 
  the 
  four 
  electrions 
  are 
  at 
  the 
  four 
  corners 
  of 
  an 
  equilateral 
  

   tetrahedron. 
  

  

  § 
  18. 
  For 
  five 
  electrions 
  we 
  have 
  clearly 
  stable 
  equilibrium 
  

   with 
  three 
  of 
  them 
  in 
  one 
  plane 
  through 
  0, 
  and 
  the 
  other 
  two 
  

   at 
  the 
  ends 
  of 
  the 
  diameter 
  perpendicular 
  to 
  this 
  plane. 
  

   There 
  is 
  also 
  at 
  least 
  one 
  other 
  configuration 
  of 
  equilibrium: 
  

   this 
  we 
  see 
  by 
  imagining 
  four 
  of 
  the 
  electrions 
  constrained 
  to 
  

   remain 
  in 
  a 
  freely 
  movable 
  plane, 
  which 
  gives 
  stable 
  equili- 
  

   brium 
  with 
  this 
  plane 
  at 
  some 
  distance 
  from 
  the 
  centre 
  and 
  

   the 
  fifth 
  electrion 
  at 
  the 
  far 
  end 
  of 
  the 
  diameter 
  perpendicular 
  

   to 
  it. 
  And 
  similarly 
  for 
  any 
  greater 
  number 
  of 
  electrions, 
  

   we 
  find 
  a 
  configuration 
  of 
  equilibrium 
  by 
  imagining 
  all 
  but 
  

   one 
  of 
  them 
  to 
  be 
  constrained 
  to 
  remain 
  in 
  a 
  freely 
  movable 
  

   plane. 
  But 
  it 
  is 
  not 
  easy, 
  without 
  calculation, 
  to 
  see, 
  at 
  all 
  

   events 
  for 
  the 
  case 
  of 
  only 
  five 
  electrions, 
  whether 
  that 
  

   equilibrium 
  would 
  be 
  stable 
  if 
  the 
  constraint 
  of 
  all 
  of 
  them 
  

   but 
  one 
  to 
  one 
  plane 
  is 
  annulled. 
  For 
  numbers 
  greater 
  than 
  

   five 
  it 
  seems 
  certain 
  that 
  that 
  equilibrium 
  is 
  unstable. 
  

  

  § 
  19. 
  For 
  six 
  we 
  have 
  a 
  configuration 
  of 
  stable 
  equilibrium 
  

   with 
  the 
  electrions 
  at 
  the 
  six 
  corners 
  of 
  a 
  regular 
  octahedron 
  ; 
  

   for 
  eight 
  at 
  the 
  corners 
  of 
  a 
  cube. 
  For 
  ten, 
  as 
  for 
  any 
  even 
  

   number, 
  we 
  should 
  have 
  two 
  configurations 
  of 
  equilibrium 
  

   (both 
  certainly 
  unstable 
  for 
  large 
  numbers) 
  with 
  two 
  halves 
  

  

  