﻿the 
  Alternate 
  Current 
  Generator. 
  55 
  

  

  electrical 
  power 
  developed 
  in 
  the 
  armature 
  circuit, 
  we 
  find 
  

   from 
  the 
  first 
  expression 
  for 
  E 
  in 
  § 
  10 
  that 
  

  

  = 
  j-{SaQai--Saia2 
  + 
  3S«2% 
  — 
  3Sa3a4 
  + 
  5Sa4a5— 
  oSasag-f 
  &c.}, 
  

  

  and 
  from 
  the 
  second 
  expression 
  for 
  E 
  that 
  

  

  Ex 
  = 
  ^SySlf^a^ 
  . 
  a, 
  . 
  ) 
  = 
  'f^S^D^SC.-/^ 
  . 
  a, 
  . 
  ) 
  

  

  as 
  D 
  sin/: 
  = 
  -^^ 
  (see 
  § 
  5) 
  

  

  = 
  the 
  heat 
  developed 
  in 
  the 
  armature 
  circuit. 
  

  

  Similarly 
  the 
  total 
  electrical 
  power 
  (R-\-7j)^ 
  developed 
  in 
  

   the 
  field 
  circuit 
  is 
  given 
  by 
  

  

  r}oiQ 
  (oni. 
  

  

  _^a^_ 
  ^j2Sai«2 
  — 
  2Sa2a3 
  + 
  4Sa3»4-4S«4ci5 
  + 
  &c.} 
  

  

  or 
  by 
  

  

  (in:^ 
  = 
  ^ 
  + 
  'f 
  lpS(r,a^ 
  . 
  «^ 
  . 
  ) 
  

  

  which 
  reduces 
  to 
  

  

  = 
  total 
  heat 
  developed 
  in 
  the 
  field 
  circuit 
  

  

  a 
  ^ 
  

   and 
  is 
  made 
  up 
  of 
  two 
  parts, 
  the 
  first 
  = 
  p 
  -J- 
  due 
  to 
  the 
  direct 
  

  

  exciting 
  current 
  and 
  the 
  . 
  second 
  = 
  ^ 
  (^2^ 
  -f 
  a^^ 
  + 
  ag^ 
  + 
  &c.) 
  

  

  due 
  to 
  the 
  induced 
  alternating 
  field-current. 
  

  

  Adding 
  the 
  first 
  expressions 
  for 
  Ex 
  and 
  (H 
  + 
  Ty)^ 
  and 
  

  

  — 
  a^ 
  

  

  cancelling 
  rj^ 
  against 
  p-j- 
  we 
  find 
  that 
  

  

  Bx 
  + 
  H^ 
  = 
  { 
  Sao^i 
  + 
  Sa^ag 
  + 
  Sa2a'3 
  + 
  Sa^sCi 
  + 
  &c. 
  } 
  . 
  

  

  