﻿948 
  On 
  a 
  Certain 
  Development 
  in 
  BesseVs 
  Functions, 
  

   Tne 
  solution 
  given 
  is 
  of 
  the 
  form 
  

  

  r^AxJjOx?') 
  sinh 
  k 
  k 
  :, 
  

   where 
  J 
  2 
  (fc 
  K 
  a) 
  = 
  0, 
  a 
  is 
  the 
  radius 
  of 
  the 
  cylinder, 
  and 
  A 
  A 
  is 
  

   so 
  chosen 
  that 
  

  

  r2* 
  x 
  A 
  A 
  cosh 
  (*a0 
  J 
  i(w) 
  —A 
  r 
  )* 
  

  

  whore 
  I 
  is 
  the 
  length 
  of 
  the 
  cylinder, 
  v 
  a 
  physical 
  constant, 
  

   and 
  /'(r) 
  the 
  shearing 
  stress 
  perpendicular 
  to 
  the 
  radius 
  on 
  

   the 
  endc 
  = 
  /. 
  The 
  value 
  given 
  for 
  Ax 
  is 
  

  

  A,= 
  

  

  \ 
  r/i^J^K^dr. 
  

  

  That 
  the 
  formula 
  is 
  in 
  error 
  is 
  seen 
  immediately 
  by 
  putting 
  

   f(r) 
  = 
  r, 
  for 
  all 
  the 
  coefficients 
  vanish. 
  This 
  corresponds 
  to 
  

   the 
  most 
  important 
  practical 
  ease, 
  and 
  gives 
  the 
  absurd 
  result 
  

   thai 
  a 
  cylinder 
  experiences 
  no 
  torsion 
  under 
  a 
  torsional 
  force 
  

   varying 
  as 
  the 
  distance 
  from 
  the 
  axis. 
  It 
  is 
  well 
  known 
  that 
  

   in 
  this 
  case 
  v 
  is 
  proportional 
  tore. 
  

  

  The 
  general 
  theorem 
  involved 
  is 
  as 
  follows. 
  The 
  develop- 
  

   ment 
  0?/(r) 
  into 
  a 
  series 
  lA 
  K 
  J„(/c 
  K 
  r) 
  where 
  J„+i(k 
  a 
  ) 
  = 
  is 
  

   not 
  possible. 
  The 
  series 
  rnnsi 
  include, 
  besides 
  the 
  Bessel's 
  

   functions, 
  the 
  initial 
  term 
  A 
  r", 
  a 
  term 
  which 
  satisfies 
  ^ 
  the 
  

   same 
  type 
  of 
  equation 
  as 
  J>». 
  The 
  correct 
  development 
  is 
  as 
  

   follows 
  : 
  — 
  

  

  /(?•) 
  = 
  A 
  r" 
  + 
  I 
  A 
  A 
  J„(ff 
  A 
  r), 
  from 
  to 
  1, 
  

  

  where 
  J/.+i(*a) 
  = 
  0, 
  

  

  Ao=(2n+2)f 
  1 
  /W* 
  ,n+1 
  ^ 
  

  

  Jo 
  

  

  The 
  initial 
  term 
  A 
  r 
  n 
  corresponds 
  to 
  the 
  zero 
  root 
  of 
  J 
  n 
  +\, 
  

   but 
  its 
  correct 
  value 
  cannot 
  be 
  obtained 
  by 
  a 
  limiting 
  process 
  

   from 
  the 
  general 
  coefficient. 
  

  

  Dini, 
  in 
  his 
  Serie 
  di 
  Fourier, 
  gives 
  the 
  theorem, 
  t 
  but 
  it 
  is 
  

   misprinted 
  so 
  that 
  the 
  first 
  term 
  appears 
  as 
  a 
  constant. 
  

   Nielsen, 
  Handbucli 
  der 
  Cylinderfunktionen, 
  misquotes 
  Dini's 
  

   misprinted 
  form, 
  w 
  r 
  hile 
  in 
  no 
  other 
  book 
  of 
  which 
  the 
  writer 
  

   is 
  aw 
  T 
  are 
  is 
  the 
  existence 
  of 
  the 
  extra 
  term 
  even 
  mentioned. 
  

  

  I 
  am, 
  Sirs, 
  

  

  Yours 
  very 
  truly, 
  

   The 
  Jefferson 
  Physical 
  Laboratory 
  P. 
  "W. 
  BrIDGMAN. 
  

  

  Cambridge, 
  Mass., 
  U.S.A. 
  

   July 
  22, 
  1908. 
  

  

  