﻿16 
  Prof. 
  H. 
  Nagaoka 
  on 
  the 
  Calculation 
  of 
  Maximum 
  

  

  Reverting 
  to 
  the 
  expression 
  (I.) 
  for 
  F, 
  we 
  see 
  that 
  (4) 
  is 
  

   equivalent 
  to 
  

  

  provided 
  z 
  can 
  be 
  expressed 
  by 
  means 
  of 
  ^-functions. 
  For 
  

   this 
  purpose 
  we 
  take 
  advantage 
  of 
  the 
  relation' 
  

  

  g 
  i~f9 
  Z. 
  ( 
  A 
  ~ 
  a) 
  2 
  +z 
  2 
  

  

  e 
  2 
  — 
  e 
  z 
  4 
  Ka 
  

  

  e 
  1 
  -e 
  3 
  = 
  (A 
  + 
  a) 
  2 
  -^z 
  2 
  

   e 
  2 
  — 
  e 
  3 
  4Aa 
  

  

  Expressed 
  by 
  means 
  of 
  ^-functions 
  

  

  * 
  2 
  ,V(0) 
  (A-a) 
  2 
  

   Aa 
  V(0) 
  Aa 
  

  

  and 
  j*_ 
  V(0) 
  (A 
  + 
  q) 
  2 
  . 
  

  

  Aa~ 
  V(0) 
  ' 
  Aa 
  

  

  Adding 
  them, 
  we 
  obtain, 
  by 
  utilizing 
  the 
  relation 
  

  

  V(0)-V(0)=V(0), 
  

  

  * 
  _ 
  g 
  / 
  V(0) 
  + 
  V(0) 
  \ 
  /A 
  , 
  a\ 
  f 
  » 
  

  

  Aa~* 
  W(0) 
  - 
  V(0)7 
  \a 
  + 
  AJ 
  ' 
  * 
  ' 
  w 
  

  

  which 
  enables 
  us 
  to 
  expand 
  z 
  in 
  terms 
  of 
  q. 
  

  

  For 
  expansion, 
  it 
  is 
  convenient 
  to 
  use 
  the 
  product 
  series 
  

   of 
  S's 
  : 
  thus 
  

  

  ^(0)=n(i-^)(i 
  + 
  g 
  2w 
  - 
  1 
  ) 
  2 
  , 
  

   V0)=n(i-^)(i-^- 
  1 
  ) 
  2 
  , 
  

  

  by 
  which 
  (1 
  — 
  q 
  2n 
  ) 
  being 
  common 
  to 
  the 
  numerator 
  and 
  

   denominator 
  is 
  eliminated. 
  On 
  evaluation 
  we 
  obtain 
  

  

  Aa 
  

  

  _l+2O? 
  2 
  -62a* 
  + 
  216g 
  6 
  -64l0 
  8 
  + 
  1636q 
  lo 
  -3778tf 
  12 
  + 
  8248tf 
  

  

  — 
  7* 
  

  

  4? 
  

  

  ... 
  (6) 
  

  

  , 
  A 
  a 
  

  

  where 
  v= 
  — 
  \- 
  -r- 
  

  

  a 
  A 
  

  

  