﻿10 
  

  

  Lord 
  Rayleigh 
  on 
  the 
  

  

  secures 
  that 
  A 
  = 
  |H 
  over 
  that 
  part 
  of 
  the 
  range 
  where 
  h=h 
  2 
  . 
  

   When 
  & 
  = 
  /*!, 
  /i— 
  |H 
  is 
  negative 
  ; 
  and 
  the 
  second 
  condition 
  

   requires 
  that 
  over 
  the 
  range 
  from 
  to 
  c 
  x 
  

  

  \h 
  zxdx 
  

  

  — 
  x 
  

  

  be 
  positive, 
  or 
  since 
  c 
  x 
  is 
  the 
  greatest 
  value 
  of 
  x 
  involved, 
  

   that 
  

  

  §h-*xdx-c^h-*dx=+. 
  . 
  . 
  . 
  (35) 
  

  

  The 
  integrals 
  can 
  be 
  written 
  down 
  at 
  once, 
  and 
  the 
  con- 
  

   dition 
  becomes 
  

  

  #'«* 
  f 
  /«A 
  (36) 
  

  

  whence 
  on 
  substitution 
  of 
  the 
  value 
  of 
  c 
  2 
  /ci 
  from 
  (34), 
  

  

  k(2k-3) 
  2 
  >l 
  (37) 
  

  

  If 
  k 
  be 
  such 
  as 
  to 
  satisfy 
  (37) 
  and 
  c 
  2 
  jc 
  x 
  be 
  then 
  chosen 
  in 
  

   accordance 
  with 
  (34) 
  and 
  regarded 
  as 
  fixed, 
  every 
  admissible 
  

   variation 
  of 
  h 
  diminishes 
  P. 
  But 
  the 
  ratio 
  c 
  2 
  /ci 
  is 
  still 
  at 
  

   disposal 
  within 
  certain 
  limits, 
  while 
  c 
  x 
  + 
  c 
  2 
  ( 
  = 
  c)iis 
  prescribed. 
  

  

  In 
  terms 
  of 
  k 
  and 
  c 
  by 
  (34) 
  

  

  Cl 
  = 
  

  

  1 
  + 
  2A 
  3 
  -3A 
  2 
  ' 
  

  

  c 
  2 
  — 
  

  

  c(2k 
  3 
  -?P) 
  

  

  (38) 
  

  

  and 
  by 
  (7) 
  

  

  . 
  . 
  . 
  (39) 
  

  

  The 
  maximum 
  of 
  f(k) 
  is 
  0*20626, 
  and 
  it 
  occurs 
  when 
  

   k=l'S7. 
  The 
  following 
  shows 
  also 
  the 
  neighbouring 
  values: 
  

  

  k. 
  

  

  1-86 
  

   1-87 
  

  

  1-88 
  

  

  0-20624 
  

   0-20626 
  

   0-20617 
  

  

  k(2k-3y 
  

   0-964 
  

   1-024 
  

   1-086 
  

  

  It 
  will 
  seen 
  that 
  while 
  A 
  = 
  1*86 
  is 
  inadmissible 
  as 
  not 
  

   satisfying 
  (37), 
  & 
  = 
  1*87 
  is 
  admissible 
  and 
  makes 
  

  

  yuUc 
  2 
  

  

  P 
  = 
  -20626' 
  

  

  w 
  

  

  (40) 
  

  

  no 
  great 
  increase 
  on 
  (18). 
  It 
  may 
  be 
  repeated 
  that 
  k 
  is 
  

  

  