﻿18 
  Prof. 
  H. 
  Nagaoka 
  on 
  the 
  Calculation 
  of 
  Maximum 
  

  

  The 
  coefficients 
  are 
  exact 
  up 
  to 
  that 
  of 
  q 
  u 
  ; 
  the 
  rest 
  of 
  

   higher 
  terms 
  are 
  insignificant 
  and 
  may 
  be 
  neglected 
  even 
  

   in 
  the 
  most 
  unfavourable 
  cases. 
  For 
  practical 
  calculation, 
  

   the 
  above 
  equation 
  is 
  applicable 
  to 
  values 
  of 
  r 
  ranging 
  from 
  

   r=oo 
  to 
  r 
  = 
  2'5, 
  the 
  latter 
  corresponding 
  to 
  the 
  case 
  of 
  

   a/A 
  =0*5. 
  From 
  this 
  value 
  down 
  tor 
  — 
  2, 
  which 
  is 
  the 
  smallest 
  

   occurring 
  in 
  practice 
  for 
  the 
  case 
  of 
  coils 
  of 
  equal 
  radii, 
  we 
  

   have 
  to 
  use 
  an 
  expression 
  in 
  terms 
  of 
  ft, 
  which 
  can 
  be 
  ex- 
  

  

  dF 
  2 
  

   pressed 
  by 
  substituting 
  (5) 
  in 
  (I.) 
  and 
  making 
  ^—=0.' 
  The 
  

  

  formula 
  is 
  as 
  follows 
  : 
  — 
  d# 
  

  

  r{l-16ft 
  + 
  376ft 
  2 
  -4672ft 
  3 
  + 
  38948ft 
  4 
  -252192ft 
  5 
  

  

  + 
  1365888ft 
  6 
  -6463360ft7 
  + 
  27500946ft 
  8 
  -. 
  . 
  . 
  . 
  

  

  - 
  120ft 
  2 
  logn 
  i 
  . 
  (1 
  - 
  16ft 
  + 
  154ft 
  2 
  - 
  1120ft 
  3 
  

  

  + 
  6680ft 
  4 
  -34272ft 
  5 
  + 
  156268ft 
  6 
  -. 
  . 
  . 
  .} 
  

   = 
  2{l-264ft 
  2 
  + 
  4096ft 
  3 
  -36828ft 
  4 
  + 
  245760ft 
  5 
  

  

  - 
  860712ft 
  6 
  + 
  6414336ft 
  7 
  -27377262ft 
  8 
  + 
  . 
  . 
  . 
  . 
  

  

  + 
  24ft 
  2 
  logn 
  - 
  . 
  (3 
  - 
  64ft 
  + 
  702ft 
  2 
  - 
  5376ft 
  3 
  

  

  + 
  32712ft 
  4 
  -169374ft 
  5 
  + 
  775908ft 
  6 
  - 
  ....)}. 
  (IF.) 
  

  

  It 
  is 
  worthy 
  of 
  remark 
  that 
  (II'.) 
  is 
  to 
  be 
  used 
  from 
  

   a/A 
  = 
  0*5 
  to 
  1, 
  the 
  value 
  of 
  ft 
  ranging 
  from 
  ft 
  = 
  0*0113 
  to 
  

   ft 
  = 
  0. 
  The 
  series 
  in 
  the 
  numerator 
  and 
  denominator 
  con- 
  

   verge 
  very 
  rapidly, 
  and 
  we 
  can 
  sometimes 
  utilize 
  the 
  formula 
  

   for 
  a 
  somewhat 
  larger 
  value 
  of 
  ft 
  ; 
  the 
  only 
  tedious 
  process 
  

  

  of 
  calculation 
  is 
  finding 
  logn 
  — 
  . 
  

  

  When 
  once 
  the 
  value 
  of 
  either 
  q 
  or 
  ft 
  is 
  found, 
  we 
  can 
  

   calculate 
  z 
  by 
  (6) 
  or 
  (6') 
  ; 
  and 
  then 
  F 
  by 
  (I.) 
  or 
  (F.). 
  

  

  As 
  an 
  example 
  of 
  practical 
  calculation, 
  let 
  us 
  take 
  the 
  case 
  

   #/A 
  = 
  0*5 
  ; 
  i. 
  e. 
  r 
  = 
  2*5. 
  From 
  the 
  first 
  three 
  terms 
  in 
  (II.), 
  

   we 
  find 
  by 
  inspection 
  that 
  q 
  is 
  nearly 
  0*11 
  ; 
  putting 
  this 
  

   value 
  in 
  (II.) 
  and 
  calculating 
  to 
  q 
  7 
  , 
  we 
  find 
  that 
  the 
  right- 
  

   hand 
  side 
  is 
  about 
  2*5059, 
  giving 
  Ar= 
  —0*0059, 
  and 
  hence 
  

   Aq 
  = 
  0*00054 
  ; 
  next 
  putting 
  q 
  = 
  0*11054, 
  we 
  find 
  

  

  Ar= 
  -0*00003476 
  

  

  by 
  taking 
  all 
  the 
  terms 
  into 
  account 
  ; 
  thus 
  the 
  final 
  value 
  

   of 
  q 
  corresponding 
  to 
  the 
  required 
  maximum 
  is 
  

  

  2= 
  0-110543224, 
  log 
  q 
  = 
  1-04353213. 
  

  

  