﻿438 
  Mr. 
  A. 
  P. 
  Wills 
  on 
  the 
  Susceptibility 
  of 
  

  

  Y,=Z 
  =laW=^9?M, 
  

  

  y 
  A.7T 
  Air 
  

  

  47T 
  47T 
  

  

  X 
  y 
  =Y 
  <B 
  = 
  ^ 
  SM=^3RL. 
  

  

  47T 
  4:7T 
  

  

  where 
  X 
  x 
  represents 
  the 
  X-component 
  of 
  the 
  force 
  per 
  unit 
  

   of 
  area 
  upon 
  a 
  plane 
  element 
  perpendicular 
  to 
  the 
  X-axis 
  (an 
  

   analogous 
  meaning 
  is 
  to 
  be 
  given 
  Y 
  y 
  , 
  Z 
  2 
  , 
  and 
  X 
  y 
  , 
  X 
  z 
  , 
  Y 
  z 
  , 
  Y*, 
  

   Zi-j 
  Z 
  y 
  ), 
  and 
  L, 
  M, 
  N, 
  £, 
  DJi, 
  9? 
  are 
  the 
  components 
  of 
  the 
  

   magnetic 
  force 
  H 
  and 
  the 
  magnetic 
  induction 
  *S 
  respectively. 
  

  

  Referring 
  to 
  fig. 
  3, 
  let 
  the 
  axis 
  of 
  X 
  be 
  in 
  the 
  line 
  of 
  

   symmetry 
  perpendicular 
  to 
  the 
  pole 
  faces 
  ; 
  let 
  the 
  axis 
  of 
  Z 
  

   be 
  vertical 
  and 
  the 
  axis 
  of 
  Y 
  be 
  perpendicular 
  to 
  the 
  axes 
  of 
  

   X 
  and 
  Z. 
  We 
  wish 
  to 
  find 
  an 
  expression 
  for 
  the 
  mechanical 
  

   force 
  acting 
  upon 
  the 
  slab 
  when 
  placed 
  relatively 
  to 
  the 
  pole- 
  

   pieces 
  as 
  shown 
  in 
  fig. 
  3. 
  There 
  will 
  be 
  discontinuity 
  in 
  the 
  

   stress-components 
  X 
  x 
  , 
  Y 
  Sf 
  Z 
  z 
  , 
  Y 
  z 
  , 
  Z 
  x 
  , 
  X 
  y 
  at 
  the 
  surface 
  of 
  the 
  

   slab 
  where 
  there 
  is 
  discontinuity 
  in 
  the 
  permeability 
  fi 
  and 
  

   the 
  forces 
  per 
  unit 
  area 
  are 
  equal 
  to 
  the 
  amounts 
  of 
  the 
  dis- 
  

   continuities. 
  Let 
  the 
  permeability 
  of 
  the 
  slab 
  be 
  /z 
  2 
  , 
  and 
  let 
  

   the 
  permeability 
  of 
  the 
  surrounding 
  medium 
  (air, 
  in 
  the 
  

   present 
  case) 
  be 
  /x 
  x 
  . 
  The 
  slab 
  used 
  is 
  so 
  thin 
  that 
  through- 
  

   out 
  its 
  whole 
  thickness 
  the 
  tubes 
  of 
  induction 
  may 
  be 
  assumed 
  

   parallel 
  to 
  the 
  X-axis, 
  and 
  consequently 
  perpendicular 
  to 
  the 
  

   faces 
  perpendicular 
  to 
  the 
  X-axis. 
  

  

  With 
  this 
  assumption 
  we 
  have 
  then 
  

  

  Z 
  2 
  = 
  Xy= 
  Y 
  z 
  =Za;='X 
  3/ 
  = 
  0, 
  

  

  and 
  ff 
  X 
  z 
  d 
  y 
  dz 
  taken 
  over 
  the 
  two 
  surfaces 
  of 
  the 
  slab 
  per- 
  

   pendicular 
  to 
  the 
  X-axis 
  equals 
  zero. 
  In 
  a 
  similar 
  way 
  the 
  

   surface-integral 
  of 
  the 
  forces 
  taken 
  over 
  the 
  two 
  surfaces 
  

   perpendicular 
  to 
  the 
  Y-axis 
  vanishes. 
  But 
  now 
  consider 
  the 
  

   forces 
  on 
  the 
  two 
  faces 
  perpendicular 
  to 
  the 
  Z-axis. 
  The 
  

   conditions 
  of 
  symmetry 
  applicable 
  to 
  the 
  other 
  faces 
  do 
  not 
  

   hold 
  here. 
  Consider 
  the 
  lower 
  Z-face, 
  that 
  is, 
  the 
  face 
  be- 
  

   tween 
  the 
  poles. 
  The 
  equations 
  show 
  that 
  all 
  forces 
  vanish 
  

   at 
  this 
  face 
  save 
  Z*, 
  and 
  we 
  have 
  then 
  the 
  force 
  Z 
  z 
  given 
  in 
  

   medium 
  of 
  permeability 
  fi 
  ± 
  by 
  

  

  1_ 
  

   Sir 
  

  

  and 
  in 
  the 
  slab 
  or 
  medium 
  of 
  permeability 
  /jl 
  2 
  by 
  

  

  Z,= 
  £ 
  : 
  » 
  1 
  H 
  1 
  , 
  

  

  07T 
  

  

  