﻿Notices 
  respecting 
  New 
  Books, 
  339 
  

  

  Chapter 
  III., 
  Mr. 
  Hardy 
  makes 
  use 
  of 
  the 
  conception 
  of 
  displace- 
  

   ment 
  in 
  a 
  plane; 
  and 
  ostensibly 
  makes 
  the 
  properties 
  of 
  the 
  

   complex 
  variable 
  depend 
  upon 
  a 
  definition 
  of 
  Multiplication 
  of 
  

   Displacements 
  (§ 
  28) 
  . 
  The 
  vector 
  addition 
  of 
  displacements 
  is 
  

   not 
  necessarily 
  in 
  a 
  plane 
  ; 
  why 
  then 
  should 
  their 
  multiplication 
  

   be 
  confined 
  to 
  a 
  plane 
  ? 
  There 
  is 
  a 
  looseness 
  of 
  argument 
  in 
  the 
  

   statement 
  that 
  "if 
  any 
  definition 
  of 
  such 
  product 
  {i. 
  e. 
  of 
  two 
  

   displacements) 
  is 
  to 
  be 
  of 
  any 
  use, 
  the 
  product 
  of 
  two 
  dis- 
  

   placements 
  must 
  itself 
  he 
  a 
  displacement" 
  Finally, 
  led 
  ostensibly 
  

   by 
  a 
  geometrical 
  relation 
  which 
  is 
  said 
  to 
  suggest 
  the 
  " 
  right 
  defi- 
  

   nition," 
  he 
  finds 
  that 
  

  

  But 
  if 
  there 
  had 
  been 
  no 
  complex 
  variable 
  to 
  guide 
  the 
  assumptions, 
  

   would 
  it 
  not 
  have 
  been 
  as 
  reasonable 
  to 
  have 
  assumed 
  

  

  [•^% 
  ^] 
  l^\ 
  y'^ 
  = 
  \.^^'+yy\ 
  ^y-y^'^'~\ 
  ? 
  

  

  This 
  indeed 
  is 
  very 
  nearly 
  Hamilton's 
  definition 
  of 
  the 
  product 
  

   of 
  two 
  vectors, 
  and 
  is 
  one 
  which 
  leads 
  to 
  spacial 
  symmetry. 
  

   However, 
  having 
  got 
  what 
  he 
  calls 
  the 
  "right 
  definition,'' 
  

   Mr. 
  Hardy 
  finds 
  it 
  "convenient 
  to 
  represent 
  the 
  pair 
  of 
  real 
  

   numbers 
  cc 
  y 
  by 
  the 
  symbol 
  x-\'iyr 
  This 
  gives 
  the 
  assumed 
  

   multiplication 
  law 
  when 
  i^= 
  — 
  1, 
  and 
  lo 
  ! 
  the 
  thing 
  is 
  done. 
  In 
  

   spite 
  of 
  the 
  footnote 
  on 
  page 
  67, 
  very 
  few 
  students 
  will 
  accept 
  

   this 
  demonstration 
  T^ithout 
  at 
  the 
  same 
  time 
  taking 
  for 
  granted 
  

   that 
  a 
  displacement, 
  a 
  vector, 
  and 
  the 
  complex 
  variable 
  are 
  names 
  

   for 
  one 
  and 
  the 
  same 
  thing. 
  By 
  calling 
  the 
  usual 
  geometrical 
  

   representation 
  of 
  the 
  complex 
  variable 
  a 
  displacement, 
  writers 
  on 
  

   function 
  theory 
  do 
  a 
  disservice 
  to 
  the 
  student 
  who 
  nine 
  cases 
  out 
  

   of 
  ten 
  will 
  continue 
  to 
  beheve 
  that 
  displacements 
  in 
  their 
  hine- 
  

   matical 
  significance 
  can 
  be 
  multiplied 
  together 
  to 
  produce 
  a 
  dis- 
  

   placement 
  — 
  a 
  statement 
  which 
  is 
  devoid 
  of 
  any 
  real 
  physical 
  

   meaning. 
  Chapter 
  III. 
  could 
  have 
  been 
  made 
  as 
  useful 
  without 
  

   any 
  reference 
  to 
  displacements 
  or 
  vectors. 
  Fortunately 
  the 
  doubtful 
  

   logic 
  does 
  not 
  in 
  the 
  least 
  affect 
  the 
  subsequent 
  use 
  of 
  the 
  complex 
  

   variable 
  in 
  its 
  true 
  analytical 
  significance. 
  

  

  The 
  first 
  book 
  on 
  our 
  list 
  is 
  much 
  more 
  advanced 
  than 
  the 
  others 
  

   and 
  appeals 
  to 
  a 
  more 
  limited 
  circle 
  of 
  students. 
  The 
  question 
  

   of 
  infinite 
  series 
  is 
  one 
  of 
  far-reaching 
  importance, 
  and 
  it 
  has 
  been 
  

   discussed 
  in 
  more 
  or 
  less 
  detail 
  by 
  several 
  of 
  our 
  recent 
  writers, 
  

   such 
  as 
  Whittaker, 
  Pierpont, 
  and 
  Carslaw. 
  And 
  now 
  from 
  the 
  

   able 
  pen 
  of 
  Professor 
  Bromwich 
  we 
  have 
  a 
  systematic 
  up-to-date 
  

   account 
  of 
  infinite 
  series 
  and 
  much 
  that 
  hinges 
  on 
  them. 
  To 
  make 
  

   the 
  argument 
  as 
  complete 
  as 
  possible 
  the 
  author 
  has 
  found 
  it 
  

   necessary 
  to 
  add 
  three 
  appendices 
  which 
  occupy 
  nearly 
  one 
  quarter 
  

   of 
  the 
  whole 
  book. 
  Appendix 
  I. 
  gives 
  an 
  arithmetic 
  treatment 
  of 
  

   convergence 
  and 
  limits 
  based 
  on 
  Dedekind's 
  definition 
  of 
  irrational 
  

   numbers, 
  and 
  might 
  have 
  been 
  incorporated 
  in 
  the 
  main 
  argument 
  

  

  