﻿44 
  Dr. 
  A. 
  C. 
  Crehore 
  on 
  the 
  Formation 
  of 
  the 
  

  

  constant, 
  ft 
  ft 
  1 
  cos 
  7, 
  gives 
  the 
  second 
  component! 
  Here 
  ft 
  

   denotes 
  the 
  ratio 
  of 
  the 
  linear 
  velocity, 
  g 
  3 
  of 
  the 
  electron, 
  e, 
  

   to 
  that 
  of 
  light, 
  e; 
  and 
  7 
  the 
  phase 
  difference 
  between 
  the 
  

   positions 
  of 
  the 
  two 
  electrons 
  in 
  their 
  respective 
  orbits. 
  

  

  Owing 
  to 
  this 
  simplification 
  it 
  has 
  been 
  possible 
  to 
  obtain 
  

   a 
  general 
  solution 
  (42) 
  for 
  the 
  average 
  value- 
  of 
  the 
  force, 
  

   between 
  two 
  electrons 
  due 
  to 
  the 
  first 
  component, 
  in 
  which; 
  

   all 
  the 
  powers 
  of 
  u 
  are 
  included, 
  since 
  the 
  law 
  of 
  the 
  series 
  of 
  

   terms 
  entering 
  into 
  the 
  force-equation 
  has 
  been 
  found. 
  

  

  The 
  instantaneous 
  value 
  of 
  the 
  force 
  perpendicular 
  to 
  00' 
  

   in 
  the 
  meridian 
  plane 
  is 
  given 
  by 
  (23), 
  and 
  the 
  average 
  value 
  

   in 
  the 
  general 
  form 
  in 
  (44). 
  The 
  instantaneous 
  force 
  along 
  

   the 
  circle 
  of 
  latitude* 
  that 
  is, 
  in 
  a 
  direction 
  perpendicular 
  to 
  

   each 
  of 
  the 
  other 
  two 
  directions, 
  is 
  .given 
  by 
  (38). 
  When 
  

   this 
  force 
  is 
  integrated 
  between 
  the 
  limit's 
  t=0 
  and 
  t 
  ■*=<*> 
  to 
  

   find 
  its 
  average 
  value, 
  the 
  result 
  gives 
  zero 
  for 
  the 
  force 
  in 
  

   this 
  direction. 
  

  

  Application 
  to 
  Atoms. 
  

  

  To 
  find 
  the 
  whole 
  force 
  between 
  two 
  atoms, 
  in 
  distinction 
  

   to 
  two 
  electrons, 
  similar 
  equations 
  are 
  written 
  down 
  for 
  each 
  

   combination 
  of 
  pairs 
  of 
  charges 
  that 
  there 
  is 
  in 
  the 
  two 
  atoms, 
  

   taking 
  one 
  charge 
  from 
  each 
  and 
  forming 
  all 
  possible 
  com- 
  

   binations 
  including 
  each 
  of 
  the 
  electrons 
  and 
  the 
  two 
  positive 
  

   spheres. 
  In 
  taking 
  the 
  sum 
  of 
  such 
  components, 
  due, 
  say, 
  

   to 
  the 
  effect 
  of 
  one 
  electron 
  in 
  the 
  one 
  atom 
  upon 
  a 
  ring 
  of 
  

   equally-spaced 
  electrons 
  in 
  the 
  other 
  atom, 
  it 
  is 
  necessary 
  to 
  

   add 
  together 
  the 
  different 
  powers 
  of 
  the 
  cosines 
  of 
  the 
  phase- 
  

   angles, 
  7, 
  between 
  the 
  single 
  electron 
  and 
  each 
  of 
  those 
  in 
  

   the 
  ring. 
  The 
  following 
  propositions 
  concerning 
  the 
  addition 
  

   of 
  the 
  various 
  powers 
  of 
  the 
  cosine 
  will 
  be 
  found 
  useful. 
  If 
  

   there 
  are 
  n 
  points 
  equally 
  spaced 
  in 
  a 
  circle, 
  then 
  the 
  sums 
  

   of 
  the 
  powers 
  of 
  the 
  cosine 
  of 
  the 
  angles, 
  measured 
  from 
  the 
  

   centre 
  of 
  the 
  circle, 
  between 
  any 
  point 
  in 
  the 
  circle 
  and 
  each 
  

   of 
  the 
  n 
  points 
  are 
  as 
  follows 
  : 
  — 
  

  

  X 
  — 
  n 
  I 
  9,'7t\ 
  

  

  2 
  cos 
  I 
  7 
  + 
  \ 
  — 
  ) 
  = 
  ; 
  except 
  for 
  n 
  = 
  l, 
  when 
  2 
  = 
  cos 
  7 
  ; 
  

   X 
  = 
  i 
  \ 
  nj 
  

  

  \—n 
  

   X 
  

  

  2 
  cos 
  2 
  1 
  7 
  + 
  \ 
  — 
  ) 
  = 
  -; 
  except 
  for 
  w=l, 
  or 
  2, 
  

  

  =1 
  A 
  n) 
  l 
  when 
  2 
  = 
  cos 
  2 
  7 
  or 
  2 
  cos 
  2 
  y; 
  

  

  COSM7 
  + 
  A,— 
  I 
  = 
  ; 
  except 
  for 
  n=l, 
  or 
  3, 
  

  

  1 
  ^ 
  n 
  / 
  when 
  -2 
  = 
  cos 
  3 
  7 
  or 
  f 
  cos 
  37 
  ; 
  ■ 
  

  

  x=zn 
  ( 
  2rr\ 
  3n 
  

  

  2 
  cos 
  4 
  1 
  7 
  + 
  \ 
  — 
  I 
  = 
  — 
  ; 
  ' 
  except 
  for 
  n= 
  1, 
  2. 
  or 
  4, 
  

  

  X 
  = 
  l 
  \ 
  n 
  ' 
  when 
  2 
  = 
  cos 
  4 
  7, 
  2 
  cos 
  4 
  7, 
  or 
  J 
  cos 
  £7 
  + 
  1 
  - 
  

  

  X 
  = 
  

  

  — 
  

  

  x= 
  

  

  