﻿72 
  Dr. 
  A. 
  C. 
  Crehore 
  on 
  the 
  Formation 
  of 
  the 
  

  

  where 
  ft 
  = 
  17*5 
  4 
  630 
  (^ 
  + 
  ^-\ 
  .... 
  (47) 
  

  

  and 
  

  

  & 
  2 
  = 
  2 
  m 
  2 
  ; 
  l 
  4 
  =Sm 
  4 
  ; 
  y=2m' 
  2 
  and 
  V=Sm' 
  4 
  . 
  (48) 
  

  

  All 
  terms 
  in 
  t; 
  -6 
  containing 
  /3 
  have 
  been 
  omitted 
  as 
  being 
  

   small 
  in 
  comparison 
  with 
  the 
  other 
  terms. 
  The 
  summation 
  

   indicated 
  in 
  (48) 
  must 
  be 
  taken 
  over 
  every 
  electron 
  in 
  the 
  

   atom 
  designated. 
  Table 
  III. 
  (p. 
  45) 
  will 
  be 
  found 
  convenient 
  

   as 
  giving 
  the 
  values 
  o£ 
  k 
  for 
  some 
  atoms 
  for 
  which 
  the 
  data 
  

   are 
  available.* 
  

  

  Equilibrium 
  Positions. 
  

  

  The 
  chart, 
  fig. 
  8, 
  represents 
  the 
  graphs 
  of 
  equations 
  (45) 
  

   and 
  (4G) 
  for 
  several 
  different 
  combinations 
  of 
  two 
  atoms 
  

   each, 
  when 
  the 
  force 
  is 
  equated 
  to 
  zero. 
  The 
  curves 
  are 
  

   given 
  in 
  terms 
  of 
  fiv 
  and 
  the 
  angle 
  of 
  latitude 
  X 
  as 
  polar 
  

   coordinates. 
  Equating 
  (45) 
  to 
  zero, 
  we 
  have 
  

  

  p 
  V 
  40-200cos 
  2 
  \ 
  + 
  Acos^ 
  (m 
  

  

  P 
  v 
  -~ 
  8-12cos 
  2 
  ~X 
  '' 
  ' 
  * 
  l 
  ; 
  

  

  and 
  from 
  (46) 
  

  

  /3V=10-17-5cos 
  2 
  \; 
  .... 
  (50) 
  

  

  whence 
  the 
  angle 
  A 
  at 
  which 
  equilibrium 
  is 
  obtained 
  is 
  

   given 
  by 
  the 
  equation 
  

  

  cos 
  2 
  X= 
  — 
  -. 
  f- 
  

  

  --21 
  

  

  10 
  10 
  " 
  10 
  

  

  ^& 
  + 
  ^ 
  • 
  <« 
  

  

  The 
  curves 
  as 
  given 
  are 
  independent 
  of 
  any 
  assumption 
  as 
  

   to 
  the 
  value 
  of 
  /3 
  ; 
  but 
  the 
  values 
  of 
  the 
  distance, 
  v, 
  and 
  of 
  

   tbe 
  force 
  are 
  dependent 
  upon 
  /3. 
  The 
  forces 
  along 
  and 
  

   perpendicular 
  to 
  r 
  for 
  a 
  particular 
  case, 
  each 
  atom 
  being 
  of 
  

   the 
  same 
  kind, 
  a 
  single 
  ring 
  of 
  three 
  electrons, 
  are 
  charted 
  

   in 
  figs. 
  9 
  and 
  10. 
  The 
  value 
  of 
  the 
  speed 
  of 
  the 
  electrons 
  

   chosen 
  is 
  1/316 
  the 
  velocity 
  of 
  light, 
  so 
  that 
  /3 
  = 
  3/316 
  and 
  

   yg 
  2 
  = 
  l0~ 
  5 
  . 
  These 
  charts 
  are 
  typical 
  of 
  all 
  atoms, 
  and 
  the 
  

   equations 
  have 
  been 
  worked 
  out 
  for 
  constant 
  values 
  of 
  the 
  

   forces, 
  varying 
  by 
  fixed 
  intervals, 
  so 
  that 
  the 
  rate 
  of 
  change 
  

   of 
  the 
  forces 
  and 
  their 
  directions 
  are 
  both 
  shown. 
  The 
  

   arrows 
  indicating 
  the 
  directions 
  of 
  the 
  forces 
  show 
  that, 
  for 
  

   a 
  small 
  displacement 
  of 
  the 
  atom 
  A' 
  from 
  the 
  position 
  of 
  

   equilibrium, 
  the 
  force 
  returns 
  the 
  atom 
  to 
  that 
  position 
  for 
  

  

  * 
  These 
  equations 
  apply 
  equally 
  well 
  to 
  an 
  atom 
  of 
  the 
  polar-system 
  

   kind 
  in 
  which 
  the 
  electrons 
  circulate 
  around 
  a 
  central 
  positive 
  charge, 
  

   but 
  the 
  radii 
  of 
  the 
  orbits 
  in 
  Table 
  III. 
  do 
  not 
  apply 
  to 
  this 
  kind 
  of 
  atom. 
  

  

  