﻿204 
  Mr. 
  H. 
  Hilton 
  : 
  A 
  Comparison 
  of 
  Various 
  

  

  Mr. 
  W. 
  Barlow 
  and 
  Mr. 
  Gr. 
  F. 
  H. 
  Smith 
  have 
  begun 
  a 
  

   similar 
  piece 
  of 
  work 
  together 
  for 
  the 
  latter 
  reason. 
  When 
  

   they 
  heard 
  that 
  I 
  had 
  independently 
  finished 
  such 
  a 
  com- 
  

   parison, 
  they 
  placed 
  my 
  work 
  side 
  by 
  side 
  with 
  their 
  own 
  to 
  

   some 
  extent, 
  and 
  found 
  that 
  we 
  agreed. 
  They 
  also 
  kindly 
  

   gave 
  me 
  some 
  useful 
  hints. 
  I 
  take 
  this 
  opportunity 
  of 
  

   thanking 
  them, 
  and 
  also 
  Prof. 
  H. 
  A. 
  Miers, 
  who 
  has 
  very 
  

   kindly 
  given 
  me 
  his 
  help 
  throughout. 
  

  

  The 
  chief 
  workers 
  on 
  the 
  subject 
  have 
  been 
  C. 
  Jordan 
  

   (Annali 
  di 
  Matematica 
  pur 
  a 
  ed 
  applicata, 
  Serie 
  2, 
  Band 
  ii. 
  

   1869); 
  L. 
  Sohncke 
  {Entivickelung 
  einer 
  Theorie 
  der 
  Krystall- 
  

   s£rwtor, 
  Leipzig, 
  1879); 
  E. 
  von 
  Fedorow 
  (Zeitschr. 
  f. 
  Kryst. 
  

   u. 
  Min. 
  xxiv. 
  1895) 
  ; 
  A. 
  Schoenflies 
  (Mathematische 
  Annalen, 
  

   xxviii., 
  xxix. 
  (1887), 
  & 
  xxxiv. 
  (1889), 
  and 
  Krystallsysteme 
  

   und 
  Krystallstructur, 
  Leipzig, 
  189 
  1) 
  ; 
  and 
  W. 
  Barlow 
  (Zeitschr. 
  

   f. 
  Kryst. 
  u. 
  Min. 
  xxiii. 
  1895). 
  (Fedorow's 
  work 
  was 
  pub- 
  

   lished 
  in 
  Russian 
  in 
  1890.) 
  

  

  The 
  notations 
  are 
  taken 
  from 
  the 
  periodicals 
  or 
  books 
  just 
  

   named, 
  and 
  compared 
  in 
  the 
  tables 
  given 
  below. 
  

  

  I 
  have 
  myself 
  carefully 
  compared 
  Barlow's 
  notation 
  with 
  

   that 
  in 
  Schoenflies' 
  book; 
  and 
  have 
  also 
  compared 
  the 
  two 
  

   notations 
  used 
  by 
  Schoenflies 
  in 
  his 
  book 
  and 
  in 
  the 
  Mathe- 
  

   matische 
  Annalen. 
  Fedorow 
  has 
  compared 
  his 
  own 
  work 
  

   with 
  that 
  of 
  Barlow* 
  and 
  of 
  Schoenflies 
  f, 
  and 
  thus 
  indirectly 
  

   has 
  compared 
  Schoenflies' 
  and 
  Barlow's 
  work. 
  If 
  there 
  was 
  

   any 
  discrepancy 
  between 
  this 
  and 
  my 
  own 
  direct 
  comparison, 
  

   I 
  have 
  investigated 
  the 
  case 
  carefully, 
  and 
  have 
  given 
  a 
  note 
  

   on 
  the 
  subject. 
  The 
  comparison 
  of 
  Schoenflies' 
  work 
  (and 
  

   hence 
  of 
  Fedorow's 
  and 
  Barlow's) 
  with 
  that 
  of 
  Jordan 
  and 
  

   Sohncke 
  is 
  that 
  given 
  by 
  Schoenflies 
  in 
  the 
  Mathematische 
  

   Annalen. 
  

  

  The 
  diagrams 
  given 
  by 
  Fedorow 
  \ 
  are 
  not 
  always 
  correct 
  ; 
  

   thus 
  e. 
  g. 
  the 
  diagrams 
  of 
  62 
  s, 
  63 
  <<?, 
  64 
  s, 
  49 
  A, 
  50 
  /*, 
  91 
  a, 
  92 
  a, 
  

   should 
  be 
  identical 
  with 
  those 
  of 
  18 
  s, 
  20 
  s, 
  21 
  s, 
  19 
  h, 
  24 
  h, 
  

   29 
  a, 
  21 
  «_, 
  respectively, 
  which 
  is 
  not 
  always 
  the 
  case 
  ; 
  in 
  

   particular, 
  as 
  Mr. 
  Smith 
  has 
  kindly 
  pointed 
  out 
  to 
  me, 
  the 
  

   diagrams 
  of 
  64 
  s 
  and 
  50 
  A 
  are 
  identical 
  and 
  both 
  wrong. 
  

   It 
  would 
  be 
  beyond 
  the 
  scope 
  of 
  this 
  paper 
  to 
  prepare 
  fresh 
  

   diagrams. 
  I 
  hope 
  to 
  do 
  this 
  on 
  some 
  future 
  occasion. 
  

  

  It 
  is 
  as 
  well 
  to 
  call 
  attention 
  to 
  the 
  fact 
  that 
  Fedorow 
  

   altered 
  his 
  notation 
  a 
  little 
  in 
  Zeitschr. 
  Kryst. 
  Min. 
  xxiv. 
  

   pp. 
  233 
  & 
  237 
  ; 
  that 
  Barlow 
  pointed 
  out 
  a 
  few 
  errors 
  in 
  

   his 
  previous 
  work 
  in 
  Zeitschr. 
  Kryst. 
  Min. 
  xxv. 
  pp. 
  86 
  & 
  87; 
  

  

  * 
  Zeitschr. 
  Kryst. 
  Min. 
  xxiv. 
  pp. 
  242-244. 
  

  

  t 
  Ibid. 
  xx. 
  pp. 
  48-61. 
  

  

  X 
  Tafeln 
  V., 
  VI., 
  Zeitschr. 
  Kryst. 
  Min. 
  xxiv. 
  

  

  