﻿46 
  Prof. 
  T. 
  E. 
  Lyle 
  on 
  the 
  Tlieonj 
  of 
  

  

  armature 
  coil, 
  fitted 
  in 
  the 
  usual 
  way 
  with 
  slip 
  rings 
  for 
  

   connexion 
  to 
  an 
  external 
  circuit 
  and 
  being 
  rotated 
  by 
  power 
  

   at 
  a 
  constant 
  angular 
  velocity 
  to 
  round 
  a 
  fixed 
  axis 
  which 
  is 
  

   perpendicular 
  to 
  its 
  own 
  axis 
  of 
  figure 
  and 
  to 
  the 
  direction 
  

   o£ 
  the 
  lines 
  of 
  force 
  of 
  F, 
  and 
  which 
  passes 
  through 
  its 
  own 
  

   centre. 
  It 
  is 
  required 
  to 
  determine 
  completely 
  the 
  currents 
  

   that 
  flow 
  in 
  both 
  A 
  and 
  F. 
  

  

  Let 
  X 
  and 
  f 
  be 
  the 
  currents 
  at 
  any 
  instant 
  in 
  A 
  and 
  F 
  

   respectively, 
  and 
  let 
  the 
  mutual 
  inductance 
  of 
  A 
  and 
  F 
  when 
  

   their 
  axes 
  are 
  coincident 
  be 
  ???, 
  and 
  hence 
  m 
  cos 
  cat 
  at 
  the 
  

   time 
  t. 
  Also 
  let 
  r 
  and 
  I 
  be 
  the 
  total 
  resistance 
  and 
  self- 
  

   inductance 
  of 
  the 
  A 
  circuit, 
  and 
  p^ 
  X 
  similar 
  quantities 
  for 
  

   the 
  F 
  circuit. 
  

  

  Then, 
  when 
  the 
  armature 
  is 
  being 
  driven 
  at 
  constant 
  

   angular 
  velocity 
  w, 
  and 
  x 
  and 
  f 
  are 
  flowing, 
  the 
  total 
  number 
  

   of 
  lines 
  linked 
  on 
  A 
  is 
  

  

  Lv 
  + 
  771^ 
  cos 
  (Ot, 
  

  

  and 
  the 
  number 
  linked 
  on 
  F 
  is 
  

  

  Xf 
  + 
  'in.c 
  cos 
  cot. 
  

  

  Hen 
  

  

  ce 
  

  

  rcc+ 
  -r 
  {Ix 
  + 
  m^ 
  cos 
  cot] 
  = 
  'j 
  

   P? 
  + 
  77 
  {>^? 
  + 
  ^^^^^' 
  cos 
  (vt\ 
  =v 
  

  

  (I-) 
  

  

  where 
  rj 
  is 
  the 
  applied 
  steady 
  e.m.f. 
  in 
  the 
  F 
  circuit. 
  

   2. 
  If 
  we 
  assume 
  as 
  the 
  solution 
  of 
  these 
  equations 
  

  

  ,u 
  = 
  ^^ 
  + 
  xi 
  sin 
  (cot 
  + 
  Ci) 
  + 
  X2 
  sin 
  (2o)t 
  4- 
  Cg) 
  + 
  .t'a 
  sin 
  (3a 
  t 
  + 
  c'3) 
  + 
  &c., 
  

  

  f 
  = 
  |5 
  + 
  f 
  1 
  sin 
  (cot 
  + 
  71) 
  + 
  f 
  2 
  sin 
  (2o)t 
  + 
  72) 
  + 
  I3 
  sin 
  {^cot 
  + 
  73) 
  + 
  &c., 
  

  

  we 
  can 
  see 
  at 
  once 
  on 
  substitution 
  that 
  /of 
  0=2?;, 
  and 
  that 
  

   cro 
  = 
  0, 
  and 
  it 
  will 
  be 
  shown 
  afterwards, 
  § 
  15, 
  that 
  when 
  

   .^0 
  = 
  then 
  fi, 
  a'2, 
  fg, 
  a-^, 
  ^5, 
  &c., 
  vanish, 
  or 
  in 
  words, 
  when 
  

   Xq 
  = 
  only 
  odd 
  harmonics 
  appear 
  in 
  x. 
  and 
  only 
  even 
  ones 
  

   in 
  f 
  . 
  Let 
  us 
  therefore 
  take 
  

  

  X 
  = 
  xi 
  sin 
  ((ot 
  + 
  cj) 
  + 
  ^-^3 
  sin 
  (ocot 
  + 
  C3) 
  + 
  x,^ 
  sin 
  (5(ct 
  + 
  c^) 
  + 
  &c. 
  "j 
  

  

  t 
  y 
  (11.) 
  

  

  f 
  = 
  f 
  + 
  ?2 
  sin 
  (2(ot 
  + 
  72) 
  + 
  f 
  4 
  sin 
  (4a)^ 
  + 
  74) 
  + 
  &c. 
  J 
  

  

  Now 
  any 
  harmonic 
  in 
  either 
  x 
  or 
  f, 
  for 
  instance 
  

   ,r 
  sin 
  (gwt 
  + 
  cO, 
  being 
  completely 
  specified 
  by 
  .i'^, 
  c^, 
  and 
  q, 
  

  

  