﻿a 
  New 
  Harmonic 
  Analyser, 
  89 
  

  

  we 
  have 
  

  

  i) 
  cos 
  kso 
  dx. 
  

  

  TTjo 
  

  

  On 
  the 
  other 
  hand, 
  if 
  n 
  is 
  the 
  number 
  of 
  an 
  element 
  of 
  the 
  

   machine 
  and 
  a 
  the 
  distance 
  between 
  any 
  two 
  elements, 
  and 
  

   the 
  distance 
  d 
  (fig. 
  1) 
  is 
  proportional 
  to 
  f(na), 
  the 
  machine 
  

   gives 
  

  

  £/(na) 
  cos 
  n6= 
  %f(x) 
  cos 
  — 
  0x, 
  

   o 
  o 
  7r 
  

  

  which 
  is 
  proportional 
  to 
  a 
  k 
  if 
  k= 
  — 
  . 
  Hence 
  to 
  obtain 
  the 
  

  

  integral, 
  the 
  lower 
  ends 
  of 
  the 
  vertical 
  rods 
  R 
  (PI. 
  XI.) 
  are 
  

   moved 
  along 
  the 
  levers 
  B 
  to 
  distances 
  proportional 
  to 
  the 
  

   ordinates 
  of 
  the 
  curve 
  y=f{nx). 
  

  

  The 
  curve 
  thus 
  obtained 
  for 
  a 
  k 
  is 
  a 
  continuous 
  function 
  of 
  k, 
  

   which 
  approximates 
  to 
  the 
  value 
  of 
  the 
  integral 
  as 
  the 
  number 
  

   of 
  elements 
  increases. 
  To 
  obtain 
  the 
  values 
  corresponding 
  

   to 
  the 
  coefficients 
  of 
  the 
  Fourier 
  series, 
  the 
  angle 
  6 
  = 
  7r, 
  or 
  

   the 
  corresponding 
  distance 
  on 
  the 
  curve, 
  is 
  divided 
  into 
  m 
  

   equal 
  parts. 
  The 
  required 
  coefficients 
  are 
  then 
  proportional 
  

   to 
  the 
  ordinates 
  erected 
  at 
  these 
  divisions. 
  

  

  Curve 
  No. 
  25 
  (PL 
  X 
  VIII.) 
  is 
  the 
  approximate 
  value 
  of 
  

   f 
  <f>(x) 
  cos 
  hx 
  dx 
  when 
  <f>(x) 
  = 
  constant 
  from 
  to 
  a 
  3 
  and 
  is 
  

  

  si 
  n 
  lea 
  

   zero 
  for 
  all 
  other 
  values. 
  The 
  exact 
  integral 
  is 
  — 
  - 
  7 
  — 
  . 
  The 
  

  

  accuracy 
  of 
  the 
  approximation 
  is 
  shown 
  by 
  the 
  following 
  

   table, 
  which 
  gives 
  the 
  observed 
  and 
  the 
  calculated 
  values 
  of 
  

   the 
  first 
  twenty 
  coefficients 
  for 
  a 
  = 
  40. 
  

  

  1 
  cos 
  hx 
  dx. 
  

  

  Jo 
  

  

  n. 
  Obs. 
  Calc. 
  A. 
  

  

  100-0 
  100*0 
  o-o 
  

  

  1 
  65-0 
  64-0 
  1-0 
  

  

  2 
  0-0 
  0-0 
  0-0 
  

  

  3 
  -200 
  -21-0 
  1-0 
  

  

  4 
  0-0 
  0-0 
  0-0 
  

  

  5 
  12-5 
  13-0 
  -0-5 
  

  

  6 
  - 
  1-5 
  0-0 
  -1-5 
  

  

  7 
  - 
  9-0 
  - 
  9-0 
  0=0 
  

  

  8 
  0-0 
  0-0 
  0-0 
  

  

  9 
  6-0 
  7-0 
  -1-0 
  

  

  10 
  - 
  2-0 
  0-0 
  -2-0 
  

  

  