﻿tlie 
  Alternate 
  Current 
  Generator. 
  63 
  

  

  19. 
  If 
  V 
  be 
  the 
  self-inductance 
  in 
  the 
  armature 
  circuit 
  

   either 
  external 
  to 
  the 
  armature 
  or 
  due 
  to 
  magnetic 
  leakage 
  

   in 
  it, 
  and 
  if 
  X' 
  be 
  a 
  similar 
  quantity 
  for 
  the 
  field 
  circuit, 
  the 
  

   equations 
  for 
  the 
  two 
  circuits 
  are 
  

  

  ,,dx 
  d 
  . 
  ^ 
  

  

  c. 
  ^ 
  fdP 
  d 
  -^ 
  

  

  where 
  r, 
  p, 
  7;, 
  d', 
  f, 
  have 
  the 
  same 
  significations 
  as 
  in 
  § 
  1. 
  

  

  Substituting 
  in 
  these 
  equations 
  from 
  § 
  18, 
  and 
  then 
  

   equating 
  separately 
  to 
  zero 
  each 
  set 
  of 
  vector 
  terms 
  of 
  the 
  

   same 
  order, 
  we 
  obtain 
  the 
  two 
  series 
  of 
  vector 
  equations, 
  

  

  9 
  

   with 
  ^Q 
  = 
  — 
  , 
  where 
  q 
  is 
  odd 
  and 
  p 
  even. 
  

  

  These 
  reduce 
  at 
  once 
  to 
  the 
  two 
  series, 
  

  

  or, 
  after 
  putting 
  a'^ 
  for 
  Gr^a^, 
  to 
  

  

  a'^-l 
  + 
  7>a^ 
  + 
  a'^+l 
  = 
  

  

  equations 
  of 
  exactly 
  the 
  same 
  form 
  as 
  those 
  for 
  the 
  simple 
  

   case 
  but 
  in 
  which 
  the 
  t 
  and 
  r 
  operators 
  are 
  now 
  given 
  by 
  

  

  9 
  TT 
  

  

  These 
  operators 
  having 
  been 
  calculated 
  from 
  known 
  data, 
  

   the 
  solution 
  for 
  a'j, 
  a'3, 
  a'5, 
  &c., 
  a^, 
  a^, 
  &c., 
  proceeds 
  exactly 
  

   as 
  in 
  the 
  simple 
  case, 
  and 
  as 
  a'l 
  = 
  Giai, 
  a'3 
  = 
  Ggag, 
  &c., 
  

   ai, 
  a-, 
  a.-,, 
  &o., 
  can 
  then 
  f)0 
  obtained. 
  

  

  