﻿the 
  Frequency 
  of 
  Alternating 
  Currents. 
  341 
  

  

  purpose 
  it 
  is 
  sufficient 
  to 
  state 
  that 
  in 
  the 
  many 
  cases 
  examined 
  

   all 
  of 
  the 
  harmonics 
  are 
  found. 
  This 
  holds 
  up 
  to 
  a 
  periodicity 
  

   of 
  1200 
  per 
  sec, 
  which 
  is 
  in 
  some 
  cases 
  the 
  36th 
  harmonic. 
  

   It 
  is 
  therefore 
  plain 
  that 
  any 
  or 
  all 
  of 
  the 
  overtones 
  may 
  he 
  

   used 
  in 
  the 
  determination 
  of 
  the 
  period 
  of 
  the 
  fundamental. 
  

   The 
  relation 
  between 
  the 
  self-induction 
  and 
  capacity 
  of 
  the 
  

   circuit 
  sometimes 
  gives 
  particular 
  prominence 
  to 
  one 
  of 
  the 
  

   overtones, 
  which 
  may 
  be 
  either 
  an 
  odd 
  or 
  an 
  even 
  harmonic. 
  

   This 
  note 
  may 
  then 
  be 
  used 
  most 
  advantageously 
  during 
  

   the 
  progress 
  of 
  the 
  experiments 
  with 
  the 
  alternating 
  currents. 
  

   The 
  period 
  of 
  the 
  note 
  can 
  be 
  easily 
  gotten 
  from 
  the 
  

   accompanying 
  tables. 
  

  

  Use 
  of 
  the 
  Tables. 
  

  

  Open 
  tubes 
  act 
  as 
  though 
  they 
  were 
  0*41 
  d 
  * 
  longer 
  than 
  

   their 
  real 
  length. 
  

  

  Then 
  the 
  first 
  length 
  of 
  the 
  tube 
  giving 
  maximum 
  resonance 
  

   will 
  be 
  Z=p-0-4l 
  rf= 
  JX-0-41 
  d, 
  

  

  where 
  Z= 
  actual 
  length 
  of 
  tube, 
  

  

  p 
  = 
  equivalent 
  length, 
  

   d 
  — 
  diameter 
  of 
  the 
  tube, 
  

   \=wave-length 
  of 
  the 
  sound. 
  

  

  Table 
  II. 
  is 
  computed 
  for 
  the 
  diameter 
  of 
  tube 
  of 
  4 
  cm. 
  

   and 
  so 
  the 
  correction 
  to 
  be 
  added 
  to 
  I 
  will 
  not 
  be 
  constant. 
  

  

  Add 
  the 
  correction 
  gotten 
  from 
  Table 
  I. 
  to 
  the 
  observed 
  

   value 
  of 
  Z, 
  and 
  in 
  the 
  column 
  for 
  the 
  temperature 
  of 
  the 
  tube 
  

   find 
  the 
  frequency 
  corresponding 
  to 
  the 
  length. 
  Divide 
  

   this 
  periodicity 
  by 
  the 
  number 
  of 
  the 
  harmonic, 
  and 
  the 
  

   periodicity 
  of 
  the 
  fundamental 
  will 
  be 
  the 
  result. 
  

  

  The 
  frequency 
  of 
  the 
  fundamental 
  will 
  be 
  the 
  greatest 
  

   common 
  divisor 
  of 
  the 
  frequencies 
  of 
  successive 
  harmonics. 
  

   Having 
  gotten 
  by 
  trial 
  the 
  number 
  of 
  the 
  harmonic 
  most 
  

   advantageous 
  to 
  use, 
  that 
  harmonic 
  alone 
  may 
  hereafter 
  be 
  

   used. 
  

  

  As 
  an 
  illustration 
  a 
  typical 
  case 
  will 
  be 
  given 
  : 
  

  

  Let 
  t 
  = 
  temperature 
  of 
  the 
  tube, 
  

  

  c= 
  correction 
  gotten 
  from 
  Table 
  I. 
  

   n 
  > 
  n 
  2y 
  n 
  3> 
  & 
  c 
  « 
  periodicity 
  of 
  fundamental, 
  second 
  har- 
  

   monic, 
  third, 
  &c. 
  

  

  d 
  = 
  5cm. 
  t 
  = 
  20°C. 
  

  

  (1) 
  Z=21«68, 
  c=2'02, 
  />=23-70, 
  w 
  p 
  = 
  360-00, 
  

  

  (2) 
  Z=18-30, 
  c=2-02, 
  /> 
  = 
  20'32, 
  r? 
  ? 
  =420'2. 
  

  

  By 
  inspection 
  No. 
  (1) 
  is 
  found 
  to 
  be 
  the 
  sixth 
  harmonic, 
  

   and 
  No. 
  (2) 
  the 
  seventh. 
  The 
  fundamental 
  which 
  is 
  the 
  

   highest 
  common 
  divisor 
  is 
  then 
  60'0. 
  

  

  * 
  Lord 
  Rayleigh, 
  ' 
  Theory 
  of 
  Sound/ 
  vol. 
  ii. 
  p. 
  295. 
  

  

  