﻿340 
  Intelligence 
  and 
  Miscellaneous 
  Articles. 
  

  

  o£ 
  the 
  early 
  part 
  of 
  the 
  book. 
  But 
  this 
  certainly 
  would 
  have 
  

   hindered 
  the 
  progress 
  of 
  the 
  average 
  student, 
  whose 
  mathematical 
  

   intuitions 
  predispose 
  him 
  to 
  take 
  a 
  good 
  deal 
  for 
  granted, 
  and 
  

   whose 
  appreciation 
  of 
  the 
  rigorous 
  arithmetical 
  foundation 
  cannot 
  

   be 
  expected 
  to 
  be 
  lively 
  until 
  the 
  conception 
  of 
  limits 
  and 
  the 
  

   tests 
  of 
  convergency 
  are 
  familiar. 
  Just 
  as 
  the 
  child 
  learns 
  the 
  

   rules 
  of 
  arithmetical 
  operations 
  long 
  before 
  he 
  understands 
  them, 
  

   so 
  it 
  is 
  throughout 
  the 
  whole 
  of 
  mathematical 
  study. 
  It 
  is 
  the 
  

   historic 
  method. 
  Appendix 
  II. 
  is 
  a 
  brief 
  discussion 
  of 
  the 
  log- 
  

   arithmic 
  and 
  exponential 
  functions, 
  the 
  former 
  being 
  defined 
  in 
  the 
  

   familiar 
  integral 
  form, 
  and 
  the 
  fundamental 
  properties 
  being 
  

   deduced 
  therefrom. 
  A 
  very 
  similar 
  section 
  will 
  be 
  found 
  in 
  

   Mr. 
  Hardy's 
  book. 
  It 
  is 
  valuable 
  as 
  giving 
  the 
  theory 
  of 
  the 
  

   logarithm 
  independent 
  of 
  the 
  series. 
  The 
  third 
  and 
  longest 
  

   Appendix 
  Mr. 
  Bromwich 
  introduced 
  so 
  that 
  he 
  might 
  be 
  able 
  to 
  

   use 
  the 
  theory 
  of 
  infinite 
  integrals 
  in 
  his 
  investigations 
  in 
  

   Chapter 
  IX. 
  on 
  Non-convergent 
  and 
  Asymptotic 
  Series. 
  This 
  

   additional 
  matter 
  has 
  the 
  single 
  disadvantage 
  of 
  making 
  the 
  book 
  

   more 
  bulky 
  ; 
  but 
  it 
  gives 
  practically 
  everything 
  that 
  is 
  needed 
  for 
  

   the 
  complete 
  study 
  of 
  the 
  subject. 
  Besides, 
  it 
  is 
  always 
  good 
  to 
  

   have 
  Mr. 
  Bromwich's 
  presentation 
  of 
  a 
  connected 
  series 
  of 
  

   theorems. 
  

  

  Both 
  Mr. 
  Bromwich 
  and 
  Mr. 
  Hardy 
  prefer 
  the 
  continental 
  nota- 
  

   tion 
  arc 
  Bm.00 
  for 
  the 
  arc 
  whose 
  sine 
  is 
  x 
  instead 
  of 
  the 
  more 
  usual 
  

   English 
  form 
  sin— 
  ^ 
  oc. 
  Notations 
  are 
  largely 
  a 
  question 
  of 
  custom. 
  

   Yet 
  the 
  latter 
  has 
  its 
  advantages 
  and 
  fits 
  well 
  in 
  with 
  functional 
  

   notation 
  generally. 
  If 
  2/=sin^, 
  then 
  sin-i 
  3/=a7. 
  On 
  the 
  other 
  

   hand, 
  smx=y 
  and 
  ^^ 
  = 
  arc 
  sin 
  i/ 
  fall 
  in 
  with 
  no 
  other 
  reciprocal 
  

   notational 
  scheme 
  in 
  recognized 
  use. 
  

  

  XLII, 
  Intelligence 
  and 
  Miscellaneous 
  Articles. 
  

   To 
  the 
  Editors 
  of 
  the 
  Philosophical 
  Magazine. 
  

  

  G-EIS'TLBMEN', 
  

  

  \X7 
  E 
  wish 
  to 
  express 
  our 
  regret 
  at 
  having 
  overlooked 
  the 
  fact 
  

   ^ 
  ' 
  that 
  one 
  of 
  Mr. 
  Yegard's 
  equations 
  is, 
  in 
  effect, 
  equivalent 
  

   to 
  one 
  of 
  ours. 
  It 
  may 
  not 
  be 
  out 
  of 
  place 
  to 
  mention 
  that 
  our 
  

   main 
  theoretical 
  results 
  are 
  obtained 
  independently 
  of 
  this 
  equa- 
  

   tion 
  ; 
  but 
  it 
  is 
  satisfactory 
  to 
  find 
  that 
  the 
  wholly 
  different 
  line 
  of 
  

   argument 
  used 
  by 
  Mr. 
  Yegard 
  leads 
  him 
  to 
  a 
  conclusion, 
  which, 
  as 
  

   far 
  as 
  it 
  is 
  applicable, 
  agrees 
  with 
  our 
  investigation. 
  

  

  Tours 
  faithfully, 
  

  

  Berkeley, 
  

   c. 
  y. 
  bueton. 
  

  

  