﻿the 
  Alternate 
  Current 
  Generator, 
  59 
  

  

  Note, 
  — 
  In 
  tbe 
  former 
  case 
  (^ 
  = 
  0) 
  the 
  t 
  operators 
  were 
  all 
  

   of 
  odd 
  and 
  the 
  t 
  ones 
  of 
  even 
  orders. 
  In 
  this 
  case 
  the 
  

   operators 
  of 
  either 
  class 
  are 
  of 
  both 
  orders. 
  

  

  The 
  translation 
  from 
  the 
  above 
  vector 
  solution 
  to 
  the 
  

   ordinary 
  sine 
  form 
  follows 
  as 
  in 
  § 
  10. 
  

  

  16. 
  In 
  the 
  preceding 
  solutions 
  the 
  magnetic 
  fluxes 
  have 
  

   been 
  assumed 
  to 
  be 
  in 
  phase 
  with 
  the 
  magnetizing 
  current- 
  

   turns, 
  and 
  so 
  iron 
  loss 
  due 
  to 
  hysteresis 
  and 
  eddv 
  currents 
  

   has 
  been 
  neglected. 
  -To 
  take 
  account 
  of 
  the 
  latter 
  the 
  

   interprc^tation 
  of 
  the 
  well-known 
  relation 
  B 
  = 
  yLtH 
  connecting 
  

   steady 
  magnetizing 
  force 
  and 
  induction 
  produced 
  has 
  to 
  be 
  

   modified. 
  

  

  The 
  induction 
  produced 
  by 
  H 
  = 
  Hi 
  sin 
  (co^ 
  + 
  Ci) 
  is 
  known 
  

   to 
  be 
  of 
  the 
  form 
  

  

  B 
  = 
  mjli 
  sin 
  (ftj^ 
  + 
  Ci 
  — 
  Si) 
  + 
  higher 
  harmonics, 
  

  

  and 
  attending 
  only 
  to 
  the 
  fundamental 
  harmonic 
  in 
  B, 
  if 
  H 
  

   be 
  represented 
  as 
  explained 
  in 
  § 
  2 
  by 
  the 
  vector 
  /ii, 
  and 
  B 
  by 
  

   the 
  vector 
  6i, 
  then 
  the 
  above 
  trigonometrical 
  relation 
  may 
  be 
  

   written 
  

  

  ^1 
  = 
  H'lh 
  

  

  where 
  yu-i 
  is 
  the 
  operator 
  mit~^i. 
  

  

  In 
  a 
  former 
  paper 
  * 
  by 
  me 
  was 
  shown 
  how 
  these 
  per- 
  

   meabiliti/ 
  operators, 
  as 
  they 
  may 
  be 
  called, 
  can 
  be 
  determined. 
  

   They 
  depend 
  on 
  the 
  character 
  of 
  the 
  iron 
  and 
  the 
  thickness 
  

   of 
  the 
  laminae, 
  on 
  the 
  amplitude 
  and 
  period 
  of 
  the 
  funda- 
  

   mental 
  harmonic 
  of 
  the 
  induction 
  oscillation 
  they 
  refer 
  to, 
  

   and 
  to 
  some 
  extent 
  on 
  the 
  wave 
  form 
  of 
  the 
  latter. 
  

  

  For 
  the 
  purposes 
  of 
  the 
  following 
  discussion 
  we 
  will 
  

   assume 
  that 
  when 
  the 
  magnetizing 
  force 
  

  

  H 
  = 
  /ii 
  + 
  /?3-f-/i5-f 
  &c. 
  

  

  produces 
  the 
  induction 
  

  

  B 
  = 
  hi-hh-hh 
  + 
  kc, 
  

  

  then 
  hi 
  = 
  fijii^ 
  h^ 
  = 
  ^Jiz^ 
  ^^b 
  — 
  l^b^h-) 
  &c., 
  where 
  the 
  /x's 
  are 
  

   operators 
  of 
  the 
  type 
  given 
  by 
  

  

  [This 
  assumption 
  as 
  regards 
  all 
  the 
  harmonics 
  of 
  B 
  but 
  

   the 
  fundamental 
  is 
  not 
  strictly 
  in 
  accordance 
  with 
  what 
  is 
  

   known 
  concerning 
  the 
  behaviour 
  of 
  laminated 
  iron 
  under 
  

   periodic 
  magnetizing 
  forces, 
  for 
  63, 
  Z>5, 
  &c.^ 
  depend, 
  at 
  any 
  

  

  * 
  " 
  Variation 
  of 
  Magnetic 
  Hystere&is 
  w 
  ith 
  Frequency," 
  Phil. 
  Mag. 
  

   Jan. 
  1905, 
  

  

  