﻿72 
  Prof. 
  T. 
  E. 
  Lyle 
  on 
  the 
  Theory 
  of 
  

  

  but 
  

  

  ( 
  ay 
  ■■}- 
  k) 
  = 
  «o^ 
  + 
  /c^ 
  + 
  2ao« 
  COS 
  ciQ/c 
  

  

  4 
  4 
  _ 
  _ 
  

  

  = 
  «o^^ 
  — 
  9 
  — 
  9^^H 
  aaecosh 
  

  

  (o'^nr 
  com 
  

  

  and 
  i£ 
  

  

  -P 
  sin 
  ^2 
  , 
  sinbg 
  , 
  sin 
  ^4 
  , 
  o 
  

  

  we 
  find 
  that 
  

   4T 
  

  

  51V3 
  ^iVg^^g 
  51V2253V4 
  

  

  fsmbi 
  , 
  -dI 
  - 
  9 
  , 
  2 
  rsin(bi 
  + 
  /t) 
  . 
  _^ 
  7I 
  -- 
  , 
  ^ 
  -0-2 
  

  

  which 
  must 
  be 
  negative 
  if 
  the 
  machine 
  runs 
  as 
  a 
  motor. 
  

  

  Now, 
  5i, 
  o-g, 
  53, 
  0-4, 
  &c., 
  sinbj, 
  sinyS2, 
  sinbs, 
  are 
  all 
  essen- 
  

   tially 
  positive 
  and 
  therefore 
  B 
  is 
  so. 
  Also 
  uq 
  and 
  ~e 
  are 
  

   essentially 
  positive. 
  So, 
  in 
  order 
  that 
  the 
  machine 
  may 
  

   work 
  as 
  a 
  motor, 
  A, 
  the 
  phase 
  angle 
  of 
  the 
  applied 
  e.m.f. 
  

   must 
  have 
  such 
  a 
  value 
  as 
  to 
  make 
  the 
  above 
  expression 
  

   for 
  T 
  negative. 
  

  

  The 
  power 
  supplied 
  by 
  the 
  source 
  e 
  being 
  the 
  mean 
  value 
  

   of 
  the 
  product 
  of 
  e 
  and 
  x, 
  that 
  is 
  of 
  e 
  and 
  aj, 
  that 
  is 
  of 
  ^ 
  

  

  and 
  iPi(aQ-\-K) 
  is 
  

  

  = 
  — 
  X— 
  ao'^sin(7t— 
  l3i)+ 
  ^- 
  i«sinbi 
  (see 
  § 
  11) 
  

  

  = 
  - 
  o 
  - 
  1 
  «osiii 
  (/i-lJi) 
  ^sinbi 
  I 
  

  

  2si 
  [_ 
  ^ 
  com 
  J 
  

  

  It 
  is 
  interesting 
  to 
  note 
  that 
  the 
  armature 
  and 
  alternating- 
  

   field 
  currents 
  which 
  flow 
  when 
  an 
  e.m.f. 
  = 
  E 
  s\a 
  (cot 
  + 
  h) 
  is 
  

   acting 
  in 
  the 
  armature 
  circuit, 
  the 
  angular 
  velocity 
  of 
  the 
  

   armature 
  is 
  o), 
  and 
  the 
  exciting 
  current 
  C, 
  would 
  be 
  un- 
  

   changed 
  if 
  the 
  E.Mir. 
  E 
  sin 
  (wi 
  + 
  /i) 
  be 
  removed, 
  the 
  speed 
  

   maintained, 
  and 
  the 
  exciting 
  field-current 
  changed 
  from 
  

  

  C 
  to 
  A 
  /C2+ 
  — 
  CE 
  cos7i+ 
  -^ 
  2E 
  

  

  This 
  follows 
  immediately 
  from 
  the 
  vector 
  equations 
  in 
  § 
  31 
  

   connecting 
  aj, 
  ag? 
  ^3? 
  &^-? 
  '^vith 
  a^ 
  + 
  zc 
  when 
  k^^ 
  /C5, 
  /tj, 
  &c., 
  

   are 
  zero. 
  

  

  2Q. 
  The 
  case 
  of 
  the 
  synchronous 
  motor 
  with 
  sinusoidal 
  

   applied 
  e.m.f., 
  discussed 
  in 
  the 
  last 
  paragraph, 
  can 
  easily 
  

   be 
  represented 
  geometrically. 
  

  

  