﻿368 
  Mr. 
  R. 
  A. 
  Lehfeldt 
  on 
  a 
  Numerical 
  Evaluation 
  of 
  

  

  in 
  that 
  form 
  in 
  the 
  equation 
  before 
  integrating 
  : 
  but 
  as 
  e 
  is 
  so 
  

   small, 
  even 
  for 
  carbon 
  dioxide, 
  and 
  is 
  known 
  with 
  so 
  little 
  

   accuracy, 
  it 
  is 
  not 
  worth 
  while 
  to 
  do 
  so 
  ; 
  it 
  is 
  sufficient 
  to 
  find 
  

   the 
  average 
  value 
  over 
  the 
  range 
  of 
  integration, 
  and 
  take 
  it 
  as 
  

   constant 
  at 
  that 
  ; 
  the 
  error 
  committed 
  is 
  less 
  than 
  the 
  errors 
  

   in 
  the 
  experimental 
  data. 
  The 
  same 
  is 
  true 
  of 
  K 
  P 
  ; 
  for 
  while 
  

   it 
  has 
  been 
  measured 
  more 
  accurately 
  than 
  e, 
  its 
  variations 
  

   are 
  less, 
  indeed 
  practically 
  nil 
  for 
  hydrogen, 
  air, 
  and 
  nitrogen. 
  

   The 
  logarithmic 
  form 
  of 
  the 
  equation 
  is 
  the 
  most 
  con- 
  

   venient 
  for 
  calculation 
  ; 
  but 
  on 
  account 
  of 
  the 
  smallness 
  of 
  

   K 
  P 
  e/v 
  we 
  find 
  approximately 
  that 
  

  

  l/T 
  =/S(l+/ 
  4 
  K 
  P 
  6/v), 
  .... 
  (vi.) 
  

  

  where 
  h 
  is 
  a 
  numerical 
  constant 
  =1*163 
  ... 
  ; 
  this 
  leads 
  to 
  

   the 
  important 
  conclusion 
  that 
  /3, 
  the 
  coefficient 
  of 
  pressure, 
  

   varies 
  linearly 
  with 
  1/v, 
  i. 
  e. 
  with 
  the 
  density. 
  So 
  far 
  as 
  it 
  is 
  

   true 
  that 
  the 
  cooling 
  effect 
  on 
  expansion 
  is 
  proportional 
  to 
  

   the 
  change 
  of 
  pressure, 
  so 
  far 
  the 
  result 
  just 
  found 
  holds 
  

   good; 
  and 
  consequently, 
  so 
  far 
  as 
  the 
  determination 
  of 
  T 
  is 
  

   concerned, 
  by 
  merely 
  assuming 
  the 
  form 
  of 
  Joule 
  and 
  Thom- 
  

   son's 
  result 
  we 
  may 
  dispense 
  with 
  its 
  numerical 
  value 
  if 
  only 
  

   we 
  know 
  the 
  rate 
  of 
  variation 
  of 
  the 
  pressure-coefficient 
  with 
  

   the 
  density 
  (i. 
  e. 
  'd/3/~dp) 
  . 
  This 
  is 
  analogous 
  to 
  the 
  deduction 
  

   made 
  by 
  Lord 
  Kelvin 
  with 
  regard 
  to 
  the 
  coefficient 
  of 
  ex- 
  

   pansion*. 
  We 
  shall 
  have 
  occasion 
  to 
  revert 
  to 
  this 
  below 
  in 
  

   considering 
  the 
  numerical 
  values 
  of 
  e. 
  

  

  Experimental 
  Data. 
  

  

  Coefficient 
  of 
  Pressure. 
  — 
  Hydrogen, 
  according 
  to 
  Ghappuisf 
  

   has 
  the 
  coefficient 
  000366254 
  for 
  a 
  pressure 
  of 
  one 
  metre 
  at 
  

   the 
  freezing-point. 
  The 
  chief 
  earlier 
  measurements 
  are 
  those 
  

   of 
  Begnault 
  0'0036678, 
  Magnus 
  0'0036594, 
  Jolly 
  0'0036562, 
  

   all 
  for 
  one 
  atmosphere 
  at 
  the 
  freezing-point. 
  It 
  is 
  impossible 
  

   to 
  draw 
  any 
  conclusions 
  from 
  these 
  numbers 
  as 
  to 
  the 
  varia- 
  

   tion 
  of 
  the 
  coefficient 
  with 
  the 
  density 
  of 
  the 
  gas, 
  a 
  variation 
  

   which 
  in 
  any 
  case 
  must 
  be 
  extremely 
  small. 
  We 
  shall 
  

   therefore 
  take 
  Chappuis's 
  result 
  simply, 
  or 
  

  

  /3 
  = 
  0'00366254. 
  

   Air. 
  — 
  Jochmann 
  J 
  and 
  Weinstein 
  § 
  in 
  evaluating 
  the 
  ther- 
  

  

  * 
  Kelvin, 
  Encycl 
  Britt. 
  art, 
  " 
  Heat." 
  

  

  t 
  P. 
  Chappuis, 
  Trav. 
  et 
  Mem. 
  de 
  la 
  comite 
  int. 
  des 
  poids 
  et 
  mesures^ 
  

   vol. 
  vi. 
  

  

  X 
  Jochmann, 
  Scklomilctis 
  Zeits. 
  v. 
  p. 
  106. 
  

  

  § 
  "Weinstein, 
  see 
  W. 
  Forster, 
  Metronomische 
  Beitrciye, 
  no. 
  3, 
  Berlin, 
  

   1881. 
  

  

  