﻿108 
  MR 
  JAMES 
  CLERK 
  MAXWELL 
  ON 
  THE 
  

  

  Cj 
  may 
  be 
  found 
  by 
  assuming 
  that 
  when 
  r=a 
  l 
  p=h 
  v 
  and 
  q 
  may 
  be 
  found 
  from 
  p 
  

   by 
  Equation 
  (21.) 
  

  

  As 
  the 
  expressions 
  thus 
  found 
  are 
  long 
  and 
  cumbrous, 
  it 
  is 
  better 
  to 
  use 
  the 
  

   following 
  approximations 
  : 
  — 
  

  

  «-(£&)? 
  (47 
  -> 
  

  

  K 
  m 
  + 
  6 
  fj.y 
  

  

  ( 
  9 
  m 
  a 
  \ 
  1 
  (r-a 
  2 
  \ 
  .... 
  

  

  P= 
  ( 
  r 
  )o-( 
  + 
  c 
  + 
  ij) 
  . 
  (48.) 
  

  

  In 
  these 
  expressions 
  a 
  is 
  half 
  the 
  depth 
  of 
  the 
  beam, 
  and 
  y 
  is 
  the 
  distance 
  of 
  

   any 
  part 
  of 
  the 
  beam 
  from 
  the 
  neutral 
  surface, 
  which 
  in 
  this 
  case 
  is 
  a 
  cylindric 
  

   surface, 
  whose 
  radius 
  is 
  c. 
  

  

  These 
  expressions 
  suppose 
  c 
  to 
  be 
  large 
  compared 
  with 
  a, 
  since 
  most 
  substances 
  

  

  break 
  when 
  - 
  exceeds 
  a 
  certain 
  small 
  quantity. 
  

  

  Let 
  b 
  be 
  the 
  breadth 
  of 
  the 
  beam, 
  then 
  the 
  force 
  with 
  which 
  the 
  beam 
  resists 
  

   flexure 
  =M. 
  

  

  by 
  a 
  =-i— 
  -f- 
  - 
  — 
  = 
  E-^— 
  . 
  . 
  . 
  (49.) 
  

   J 
  x 
  m+Q 
  ac 
  6 
  3c 
  v 
  J 
  

  

  M- 
  

  

  which 
  is 
  the 
  ordinary 
  expression 
  for 
  the 
  stiffness 
  of 
  a 
  rectangular 
  beam. 
  

  

  The 
  stiffness 
  of 
  a 
  beam 
  of 
  any 
  section, 
  the 
  form 
  of 
  which 
  is 
  expressed 
  by 
  an 
  

   equation 
  between 
  x 
  and 
  y, 
  the 
  axis 
  of 
  x 
  being 
  perpendicular 
  to 
  the 
  plane 
  of 
  flexure, 
  

   or 
  the 
  osculating 
  plane 
  of 
  the 
  axis 
  of 
  the 
  beam 
  at 
  any 
  point, 
  is 
  expressed 
  by 
  

  

  Mc 
  

  

  = 
  Kjifdx, 
  . 
  . 
  . 
  (50.) 
  

  

  M 
  being 
  the 
  moment 
  of 
  the 
  force 
  which 
  bends 
  the 
  beam, 
  and 
  c 
  the 
  radius 
  of 
  the 
  

   circle 
  into 
  which 
  it 
  is 
  bent. 
  

  

  Case 
  VI. 
  

  

  At 
  the 
  meeting 
  of 
  the 
  British 
  Association 
  in 
  1839, 
  Mr 
  James 
  Nasmyth 
  de- 
  

   scribed 
  his 
  method 
  of 
  making 
  concave 
  specula 
  of 
  silvered 
  glass 
  by 
  bending. 
  

  

  A 
  circular 
  piece 
  of 
  silvered 
  plate-glass 
  was 
  cemented 
  to 
  the 
  opening 
  of 
  an 
  iron 
  

   vessel, 
  from 
  which 
  the 
  air 
  was 
  afterwards 
  exhausted. 
  The 
  mirror 
  then 
  became 
  

   concave, 
  and 
  the 
  focal 
  distance 
  depended 
  on 
  the 
  pressure 
  of 
  the 
  air. 
  

  

  Buffon 
  proposed 
  to 
  make 
  burning-mirrors 
  in 
  this 
  way, 
  and 
  to 
  produce 
  the 
  

   partial 
  vacuum 
  by 
  the 
  combustion 
  of 
  the 
  air 
  in 
  the 
  vessel, 
  which 
  was 
  to 
  be 
  

   effected 
  by 
  igniting 
  sulphur 
  in 
  the 
  interior 
  of 
  the 
  vessel 
  by 
  means 
  of 
  a 
  burning- 
  

   glass. 
  Although 
  sulphur 
  evidently 
  would 
  not 
  answer 
  for 
  this 
  purpose, 
  phos- 
  

   phorus 
  might 
  ; 
  but 
  the 
  simplest 
  way 
  of 
  removing 
  the 
  air 
  is 
  by 
  means 
  of 
  the 
  air- 
  

   pump. 
  The 
  mirrors 
  which 
  were 
  actually 
  made 
  by 
  Buffon, 
  were 
  bent 
  by 
  means 
  

   of 
  a 
  screw 
  acting 
  on 
  the 
  centre 
  of 
  the 
  glass. 
  

  

  