﻿Dr. 
  C. 
  Chree's 
  Notes 
  on 
  Thermometry. 
  207 
  

  

  divisions. 
  If 
  then 
  v 
  be 
  the 
  volume 
  at 
  0° 
  0. 
  of 
  one 
  

   stem-division 
  we 
  have, 
  referring 
  to 
  (1) 
  and 
  (2), 
  

  

  V 
  (l 
  + 
  A 
  1 
  * 
  + 
  A 
  2 
  * 
  2 
  + 
  ...) 
  = 
  {V 
  + 
  v 
  Q 
  (t 
  + 
  x)}{l 
  + 
  a 
  1 
  t 
  + 
  a 
  2 
  t*+ 
  ...). 
  (3) 
  

   When 
  £ 
  = 
  100°, 
  x 
  vanishes, 
  thus 
  

  

  r 
  /V 
  =(*i 
  + 
  100^ 
  2 
  + 
  100 
  2 
  6 
  3 
  + 
  ...)^(l 
  + 
  100« 
  1 
  + 
  100 
  2 
  a 
  2 
  +...) 
  J 
  . 
  (4) 
  

  

  where 
  e 
  1 
  =A 
  l 
  — 
  a 
  1 
  , 
  e 
  2 
  =A 
  2 
  — 
  a 
  2 
  , 
  &c. 
  

  

  If 
  both 
  glass 
  and 
  mercury 
  had 
  a 
  linear 
  law 
  of 
  expansion, 
  

   i. 
  e. 
  if 
  all 
  the 
  constants 
  except 
  A 
  1 
  and 
  a 
  x 
  were 
  zero, 
  (4) 
  would 
  

   become 
  

  

  Vo/V 
  =^ 
  1 
  -T-(l 
  + 
  100a 
  1 
  ), 
  (5) 
  

  

  giving 
  as 
  first 
  approximation 
  

  

  Substituting 
  from 
  (4) 
  in 
  (3) 
  and 
  reducing, 
  we 
  get 
  

   = 
  i 
  , 
  * 
  ( 
  , 
  1( 
  ?°~^, 
  [ei{ai 
  + 
  a 
  2 
  {100 
  + 
  t)+a 
  3 
  (100* 
  + 
  100t 
  + 
  t*)+...} 
  

  

  1 
  + 
  till 
  -\- 
  cc 
  2 
  t 
  + 
  . 
  . 
  . 
  

   -e 
  2 
  {l-100 
  a 
  2 
  t... 
  } 
  -* 
  3 
  {100 
  + 
  £+...} 
  + 
  ...] 
  -h(>i 
  + 
  100 
  e 
  2 
  + 
  100% 
  + 
  ...). 
  (6) 
  

  

  The 
  preceding 
  formulae 
  are 
  not 
  of 
  course 
  legitimate 
  for 
  all 
  

   values 
  of 
  t. 
  The 
  freezing- 
  and 
  boiling-points 
  of 
  mercury 
  

   assign 
  limits 
  to 
  the 
  application 
  of 
  (3), 
  and 
  in 
  the 
  immediate 
  

   neighbourhood 
  of 
  these 
  points 
  — 
  especially 
  of 
  the 
  boiling- 
  

   point 
  — 
  its 
  accuracy 
  is 
  somewhat 
  uncertain. 
  At 
  ordinary 
  

   temperatures 
  the 
  series 
  on 
  the 
  right 
  of 
  (1) 
  converges 
  rapidly, 
  

   A^ 
  being 
  much 
  the 
  most 
  important 
  term. 
  The 
  glass 
  of 
  our 
  

   present 
  problem 
  is 
  ideal. 
  There 
  is 
  perhaps 
  no 
  actual 
  glass 
  of 
  

   whose 
  expansion 
  we 
  know 
  enough 
  to 
  judge 
  whether 
  it 
  is 
  

   expedient 
  to 
  retain 
  any 
  term 
  higher 
  than 
  a 
  2 
  t 
  2 
  in 
  (2) 
  . 
  In 
  all 
  

   ordinary 
  kinds 
  of 
  glass 
  cti/Ai 
  is 
  of 
  the 
  order 
  1/7, 
  and 
  so 
  e 
  1 
  

   may 
  be 
  assumed 
  necessarily 
  positive. 
  The 
  sign 
  of 
  e 
  2 
  is 
  more 
  

   uncertain, 
  and 
  as 
  regards 
  e% 
  and 
  e± 
  nothing 
  is 
  known 
  except 
  

   that 
  the 
  terms 
  depending 
  on 
  them 
  appear 
  to 
  be 
  of 
  little 
  

   importance. 
  

  

  {Supposing 
  our 
  ideal 
  glass 
  to 
  resemble 
  ordinary 
  glass 
  in 
  

   the 
  general 
  laws 
  of 
  its 
  expansion, 
  (\ 
  + 
  a\b 
  + 
  a 
  2 
  t 
  2 
  + 
  . 
  . 
  .) 
  -1 
  is 
  

   nearly 
  unity 
  and 
  replaceable 
  by 
  a 
  factor 
  (1 
  — 
  ait...), 
  pro- 
  

   ceeding 
  in 
  ascending 
  powers 
  of 
  t. 
  Thus 
  (6) 
  may 
  be 
  regarded 
  

   as 
  of 
  the 
  general 
  type 
  * 
  

  

  a; 
  = 
  *(100-0( 
  B 
  o+'Bi* 
  + 
  B 
  a 
  * 
  9 
  +...), 
  . 
  . 
  . 
  (7) 
  

  

  * 
  Cf. 
  Gnillaume's 
  Thermometrie 
  . 
  . 
  ., 
  p. 
  195. 
  

   Q2 
  

  

  