﻿GO 
  Prof. 
  T. 
  R. 
  Lyle 
  on 
  the 
  Theory 
  of 
  

  

  rate 
  for 
  high 
  values 
  of 
  hi, 
  more 
  on 
  h-i 
  than 
  on 
  /is, 
  /ig, 
  &c. 
  At 
  

   the 
  same 
  time 
  it 
  is 
  hoped 
  that 
  the 
  following 
  discussion 
  may 
  

   be 
  of 
  some 
  value.] 
  

  

  In 
  general 
  if 
  H=XAg 
  produce 
  B 
  = 
  S/>o) 
  ^s 
  the 
  total 
  iron 
  

   loss 
  per 
  c.c. 
  per 
  cycle 
  due 
  to 
  both 
  hysteresis 
  and 
  eddy 
  

   currents 
  is 
  

  

  47rJ 
  47rJ^j 
  dt 
  ' 
  

  

  where 
  T 
  is 
  the 
  period, 
  the 
  total 
  iron 
  loss 
  per 
  c.c. 
  per 
  second 
  

   is 
  

  

  = 
  -j— 
  . 
  Average 
  value 
  of 
  product 
  H 
  --rr 
  

  

  TT 
  

  

  J—. 
  Av. 
  product 
  ^hg 
  into 
  t^Xqcohq 
  

  

  (See 
  § 
  11) 
  

  

  = 
  — 
  {SMi 
  + 
  3SMs 
  + 
  5Syi5 
  + 
  &c.} 
  

  

  OTT 
  

  

  Again, 
  it 
  is 
  well 
  known 
  that 
  if 
  the 
  steady 
  magnetizing 
  

   current-turns 
  nx 
  act 
  on 
  a 
  magnetic 
  circuit 
  composed 
  of 
  

   different 
  materials, 
  the 
  flux 
  F 
  produced 
  is 
  given 
  by 
  

  

  F 
  = 
  

  

  

  where 
  the 
  L, 
  A's 
  are 
  the 
  lengths 
  and 
  sectional 
  areas 
  of 
  the 
  

   different 
  portions 
  of 
  the 
  circuit, 
  and 
  the 
  fis 
  are 
  the 
  perme- 
  

   abilities 
  of 
  these 
  portions 
  for 
  the 
  particular 
  flux 
  densities 
  in 
  

   them. 
  

  

  If 
  now 
  the 
  magnetizing 
  current 
  be 
  an 
  alternating 
  one, 
  

   that 
  is 
  if 
  X 
  =^ 
  x 
  I 
  sin 
  ( 
  cot 
  + 
  Ci) 
  — 
  Sii 
  (a 
  vector) 
  the 
  same 
  equation 
  

   will 
  give 
  the 
  corresponding 
  harmonic 
  of 
  the 
  flux 
  produced, 
  

   but 
  the 
  fis 
  are 
  now 
  the 
  permeability 
  operators, 
  for 
  the 
  

   different 
  portions 
  of 
  the 
  circuit. 
  

  

  X 
  -T 
  — 
  can, 
  by 
  the 
  addition 
  theorem 
  in 
  § 
  3, 
  be 
  reduced 
  to 
  a 
  

  

  single 
  operator 
  so 
  that 
  if 
  the 
  flux/i 
  (vector) 
  be 
  produced 
  in 
  

   any 
  magnetic 
  circuit 
  by 
  the 
  current-turns 
  ^^a^ 
  we 
  have 
  

   always 
  a 
  relation 
  of 
  the 
  form 
  

  

  /i 
  = 
  Gl-i?2ai 
  

  

  where 
  Gi 
  is 
  an 
  operator 
  of 
  the 
  form 
  git 
  which 
  can 
  be 
  

   determined. 
  

  

  