﻿128 
  Dr. 
  J. 
  G-. 
  Leathern 
  on 
  the 
  

  

  Hence 
  the 
  subtraction 
  of 
  (ifc 
  2 
  co\/27r) 
  exp 
  (47rf/\) 
  from 
  

   the 
  right-hand 
  side 
  of 
  formula 
  (21) 
  gives 
  an 
  expression 
  

   which 
  has 
  a 
  definite 
  limit 
  for 
  £-><x>. 
  This 
  justifies 
  the 
  

   definition 
  

  

  v 
  2 
  + 
  iu 
  2 
  =~ 
  Lim 
  

  

  W^ 
  2 
  e 
  *nt/k 
  

  

  XC 
  W)}2 
  { 
  C 
  ^ 
  

  

  10. 
  Limiting 
  form 
  of 
  the 
  second 
  motion 
  at 
  infinity, 
  — 
  The 
  

   formula 
  (25) 
  defines 
  a 
  motion 
  which 
  has 
  all 
  the 
  characteristics 
  

   required 
  for 
  the 
  second 
  motion, 
  with 
  the, 
  as 
  yet, 
  possible 
  

   exception 
  of 
  tending 
  to 
  the 
  proper 
  form 
  at 
  infinity. 
  It 
  

   is 
  now 
  necessary 
  to 
  inquire 
  what 
  are 
  the 
  limiting 
  forms 
  to 
  

   which 
  u 
  2 
  and 
  v 
  2 
  , 
  functions 
  of 
  £ 
  and 
  tj, 
  tend 
  with 
  indefinite 
  

   increase 
  of 
  77. 
  

  

  For 
  this 
  purpose 
  the 
  series-expansions 
  of 
  formulae 
  (22) 
  

   and 
  (23) 
  may 
  be 
  used, 
  each 
  within 
  the 
  appropriate 
  range 
  

   of 
  7]'. 
  If 
  [22], 
  [23] 
  be 
  used 
  as 
  abbreviations 
  for 
  the 
  

   expressions 
  on 
  the 
  right-hand 
  side 
  of 
  these 
  formulse, 
  

  

  v 
  2 
  + 
  iu 
  2 
  = 
  ~ 
  Lim 
  I" 
  -^V^+ 
  P 
  fV($')} 
  2 
  [22] 
  d?d 
  v 
  r 
  

  

  +j 
  1 
  o 
  , 
  JW')} 
  2 
  [23]dfcV].(26) 
  

  

  As 
  the 
  integral 
  of 
  formula 
  (25) 
  is 
  absolutely 
  convergent 
  

   in 
  respect 
  of 
  the 
  infinity 
  of 
  the 
  subject 
  of 
  integration 
  

   at 
  £' 
  = 
  £, 
  it 
  is 
  safe 
  to 
  use 
  the 
  series 
  [22] 
  and 
  [23] 
  right 
  up 
  

   to 
  the 
  critical 
  value 
  77' 
  = 
  rj 
  which 
  separates 
  the 
  ranges 
  within 
  

   which 
  they 
  are 
  respectively 
  valid. 
  For 
  {h(£')} 
  2 
  the 
  series 
  

   of 
  formula 
  (24) 
  is 
  again 
  employed. 
  

  

  In 
  taking 
  the 
  term-by-term 
  products 
  of 
  the 
  two 
  series 
  

   which 
  are 
  multiplied 
  under 
  the 
  sign 
  of 
  integration, 
  any 
  

   resulting 
  term 
  may 
  be 
  passed 
  over 
  whose 
  integral 
  with 
  

   respect 
  to 
  f 
  over 
  a 
  range 
  X 
  is 
  zero. 
  Thus 
  a 
  term 
  of 
  the 
  

   the 
  type 
  

  

  cos 
  { 
  (2irm/\)(? 
  + 
  a)} 
  cos 
  { 
  (2ti7i/\) 
  [£' 
  + 
  /5) 
  } 
  

  

  need 
  not 
  be 
  considered 
  unless 
  m 
  = 
  n. 
  Further, 
  when 
  only 
  

   an 
  approximation 
  for 
  77 
  great 
  is 
  desired, 
  an 
  estimate 
  of 
  the 
  

   importance 
  of 
  an 
  exponential 
  in 
  77' 
  and 
  77 
  is 
  to 
  be 
  made 
  

   on 
  the 
  hypothesis 
  that 
  77 
  is 
  very 
  great 
  but 
  that 
  77' 
  is 
  of 
  

  

  