﻿88 
  Messrs. 
  Michelson 
  and 
  Stratton 
  on 
  

  

  About 
  a 
  year 
  ago 
  a 
  machine 
  was 
  constructed 
  on 
  this 
  prin- 
  

   ciple 
  with 
  twenty 
  elements, 
  and 
  the 
  results 
  obtained* 
  were 
  so 
  

   encouraging 
  that 
  it 
  was 
  decided 
  to 
  apply 
  to 
  the 
  Bache 
  Fund 
  

   for 
  assistance. 
  The 
  present 
  machine 
  of 
  eighty 
  elements 
  was 
  

   constructed 
  with 
  the 
  aid 
  of 
  a 
  grant 
  from 
  this 
  fund. 
  

  

  Fig. 
  1 
  shows 
  the 
  essential 
  parts 
  of 
  a 
  single 
  element, 
  s 
  is 
  

   one 
  of 
  the 
  eighty 
  small 
  springs 
  attached 
  side 
  by 
  side 
  to 
  the 
  

   lever 
  C, 
  which 
  for 
  greater 
  rigidity 
  has 
  the 
  form 
  of 
  a 
  hollow 
  

   cylinder, 
  pivoted 
  on 
  knife-edges 
  at 
  its 
  centre. 
  S 
  is 
  the 
  large 
  

   counter-spring. 
  The 
  harmonic 
  motion 
  produced 
  by 
  the 
  

   eccentric 
  A 
  is 
  communicated 
  to 
  x 
  by 
  the 
  rod 
  P 
  and 
  lever 
  B, 
  

   the 
  amplitude 
  of 
  the 
  motion 
  at 
  x 
  depending 
  on 
  the 
  adjustable 
  

   distance 
  d. 
  The 
  resultant 
  motion 
  is 
  recorded 
  by 
  a 
  pen 
  con- 
  

   nected 
  with 
  u 
  by 
  a 
  fine 
  wire 
  ; 
  under 
  the 
  pen 
  a 
  slide 
  moves 
  

   with 
  a 
  speed 
  proportional 
  to 
  the 
  angular 
  motion 
  of 
  the 
  cone 
  D 
  

   (Plate 
  XI.). 
  

  

  To 
  represent 
  the 
  succession 
  of 
  terms 
  of 
  a 
  Fourier 
  series 
  the 
  

   eccentrics 
  have 
  periods 
  increasing 
  in 
  regular 
  succession 
  from 
  

   one 
  to 
  eighty. 
  This 
  is 
  accomplished 
  by 
  gearing 
  to 
  each 
  

   eccentric 
  a 
  wheel 
  the 
  number 
  of 
  whose 
  teeth 
  is 
  in 
  the 
  proper 
  

   ratio. 
  These 
  last 
  are 
  all 
  fastened 
  together 
  on 
  the 
  same 
  axis 
  

   and 
  form 
  the 
  cone 
  D 
  (PL 
  XL). 
  

  

  Turning 
  the 
  cone 
  will 
  produce 
  at 
  the 
  point 
  x 
  motions 
  cor- 
  

   responding 
  to 
  cos 
  6, 
  cos 
  26, 
  cos 
  36, 
  &c, 
  up 
  to. 
  cos 
  800, 
  with 
  

   amplitudes 
  depending 
  on 
  the 
  distances 
  d. 
  The 
  motion 
  of 
  

   the 
  elements 
  may 
  also 
  be 
  changed 
  from 
  sine 
  to 
  cosine 
  by 
  dis- 
  

   engaging 
  the 
  cone 
  and 
  turning 
  all 
  of 
  the 
  eccentrics 
  through 
  

   90° 
  by 
  means 
  of 
  a 
  long 
  pinion, 
  which 
  can 
  be 
  thrown 
  in 
  gear 
  

   with 
  all 
  of 
  the 
  eccentric 
  wheels 
  at 
  once. 
  

  

  The 
  efficiency 
  and 
  accuracy 
  of 
  the 
  machine 
  is 
  well 
  illus- 
  

   trated 
  by 
  an 
  inspection 
  of 
  the 
  summations 
  of 
  Fourier 
  series 
  

   shown 
  by 
  curves 
  1 
  to 
  12 
  (Pis. 
  XII. 
  to 
  XIV.). 
  

  

  Curves 
  13 
  to 
  24 
  (Plates 
  XIY. 
  to 
  XVI.) 
  are 
  added 
  to 
  

   illustrate 
  the 
  use 
  of 
  the 
  machine 
  in 
  constructing 
  curves 
  repre- 
  

   senting 
  functions 
  which 
  scarcely 
  admit 
  of 
  other 
  analytical 
  

   expression. 
  

  

  PI. 
  XVII. 
  shows 
  well 
  the 
  dependence 
  of 
  the 
  accurac} 
  r 
  of 
  

   the 
  representation 
  of 
  a 
  particular 
  function 
  on 
  the 
  number 
  of 
  

   terms 
  of 
  the 
  series. 
  

  

  The 
  machine 
  is 
  capable 
  not 
  only 
  of 
  summing 
  up 
  any 
  given 
  

   trigonometric 
  series, 
  but 
  can 
  also 
  perform 
  the 
  inverse 
  process 
  

   of 
  finding 
  for 
  any 
  given 
  function 
  the 
  coefficients 
  of 
  the 
  corre- 
  

   sponding 
  Fourier 
  series. 
  Thus 
  if 
  

  

  f[x) 
  = 
  a 
  +' 
  a 
  1 
  pos 
  x 
  + 
  a 
  2 
  cos 
  2x 
  + 
  , 
  

  

  * 
  Paper 
  read 
  before 
  thc-National 
  Academy 
  of 
  Sciences, 
  April 
  1897. 
  

  

  