﻿a 
  .New 
  Harmonic 
  Analyser. 
  87 
  

  

  of 
  the 
  stretch 
  of 
  the 
  cord 
  and 
  its 
  imperfect 
  flexibility 
  ; 
  so 
  that 
  

   with 
  a 
  considerable 
  increase 
  in 
  the 
  number 
  of 
  elements, 
  the 
  

   accumulated 
  errors 
  due 
  to 
  these 
  causes 
  would 
  soon 
  neutralize 
  

   the 
  advantages 
  of 
  the 
  increased 
  number 
  of 
  terms 
  in 
  the 
  series. 
  

   It 
  occurred 
  to 
  one 
  of 
  us 
  some 
  years 
  ago 
  that 
  the 
  quantity 
  

   to 
  be 
  operated 
  upon 
  might 
  be 
  varied 
  almost 
  indefinitely, 
  and 
  

   that 
  most 
  of 
  the 
  imperfections 
  in 
  existing 
  machines 
  might 
  be 
  

   practically 
  eliminated. 
  Among 
  the 
  methods 
  which 
  appeared 
  

   most 
  promising 
  were 
  the 
  following 
  : 
  — 
  addition 
  of 
  fluid 
  pres- 
  

   sures, 
  elastic 
  and 
  other 
  forces, 
  and 
  electric 
  currents. 
  Of 
  these 
  

   the 
  simplest 
  in 
  practice 
  is 
  doubtless 
  the 
  addition 
  of 
  the 
  forces 
  

   of 
  spiral 
  springs. 
  

  

  The 
  principle 
  upon 
  which 
  the 
  use 
  of 
  springs 
  depends 
  may 
  

   be 
  demonstrated 
  as 
  follows 
  : 
  — 
  

  

  Let 
  a 
  (fig. 
  1) 
  = 
  lever-arm 
  of 
  small 
  springs 
  (but 
  one 
  of 
  

   which 
  is 
  shown 
  in 
  the 
  figure), 
  

   b 
  = 
  lever-arm 
  of 
  large 
  counter-spring 
  S, 
  

   / 
  = 
  natural 
  length 
  of 
  small 
  springs, 
  

   L 
  = 
  „ 
  „ 
  large 
  spring, 
  

  

  I 
  + 
  x 
  = 
  stretched 
  length 
  of 
  small 
  springs, 
  

   L 
  + 
  ?/= 
  „ 
  „ 
  large 
  spring, 
  

  

  e 
  = 
  constant 
  of 
  small 
  springs, 
  

   E 
  = 
  „ 
  large 
  spring, 
  

  

  n 
  = 
  number 
  of 
  small 
  springs, 
  

   p= 
  force 
  due 
  to 
  one 
  of 
  the 
  small 
  springs, 
  

   P= 
  » 
  » 
  large 
  spring; 
  

  

  then 
  

  

  / 
  

  

  whence 
  

  

  e 
  ( 
  . 
  a 
  \ 
  

  

  p= 
  T\ 
  l+x 
  -b 
  y 
  

  

  a%p 
  = 
  bP, 
  

  

  y 
  

  

  i 
  

  

  n' 
  

  

  <i+i) 
  

  

  From 
  this 
  it 
  follows 
  that 
  the 
  resultant 
  motion 
  is 
  propor- 
  

   tional 
  to 
  the 
  algebraic 
  sum 
  of 
  the 
  components, 
  at 
  least 
  to 
  the 
  

   same 
  order 
  of 
  accuracy 
  as 
  the 
  increment 
  of 
  force 
  of 
  every 
  

   spring 
  is 
  proportional 
  to 
  the 
  increment 
  of 
  length. 
  

  

  To 
  obtain 
  the 
  greatest 
  amplitude 
  for 
  a 
  given 
  number 
  of 
  

   elements 
  the 
  ratios 
  //L 
  and 
  a/b 
  should 
  be 
  as 
  small 
  as 
  possible, 
  

   but 
  of 
  course 
  a 
  limit 
  is 
  soon 
  reached 
  when 
  other 
  considerations 
  

   enter. 
  

  

  