﻿the 
  AUernate 
  Current 
  Generator, 
  47 
  

  

  can, 
  when 
  its 
  order 
  q 
  is 
  known, 
  be 
  represented 
  by 
  the 
  vector 
  

   drawn 
  from 
  the 
  origin 
  in 
  any 
  reference 
  plane 
  to 
  the 
  point 
  in 
  

   that 
  piano 
  whose 
  polar 
  coordinates 
  are 
  x^, 
  Cq, 
  twice 
  the 
  

   constant 
  term 
  in 
  f 
  being 
  represented 
  in 
  the 
  same 
  plane 
  by 
  

  

  TT 
  

  

  the 
  vector 
  to 
  the 
  point 
  f 
  o? 
  ^ 
  

  

  The 
  form 
  of 
  solution 
  (II.) 
  assumed 
  may 
  now 
  be 
  written 
  

  

  ,v 
  = 
  ai 
  + 
  ag 
  -f 
  as 
  4- 
  &c. 
  

  

  ? 
  "0^ 
  ^ 
  ^ 
  ^. 
  ^ 
  • 
  • 
  • 
  (HI.) 
  

  

  f^ 
  = 
  -^ 
  + 
  «o 
  + 
  ^4 
  + 
  as 
  + 
  cfcc- 
  

  

  where 
  ai, 
  ag, 
  &c., 
  ccq, 
  olc^, 
  ^4, 
  &c., 
  are 
  vectors 
  whose 
  orders 
  are 
  

   indicated 
  by 
  the 
  subscribed 
  numbers. 
  Of 
  these, 
  one 
  only, 
  

   namely 
  wq, 
  is 
  known, 
  as 
  it 
  is 
  drawn 
  to 
  the 
  point 
  whose 
  polar 
  

  

  co-ordinates 
  are 
  ^q? 
  t? 
  where 
  fo 
  = 
  '~' 
  The 
  others 
  have 
  to 
  

   'I 
  p 
  

  

  be 
  determined. 
  

  

  2\^ote 
  a. 
  — 
  In 
  the 
  sequel 
  it 
  will 
  sometimes 
  happen 
  that 
  a 
  

  

  vector, 
  say 
  a 
  , 
  originally 
  assumed 
  of 
  order 
  q, 
  will 
  be 
  used 
  to 
  

  

  represent 
  an 
  harmonic 
  of 
  a 
  different 
  order, 
  say 
  q 
  + 
  1. 
  In 
  such 
  

  

  a 
  case 
  it 
  will 
  be 
  written 
  (a^)^a.] 
  ; 
  thus 
  

  

  aj 
  = 
  Xq 
  sin 
  (q(ot-\-Cq), 
  

   but 
  

  

  (^q)g+l 
  = 
  ^^'q^^^ 
  {(q+l)o)ti-Cq\. 
  

  

  Note 
  b. 
  — 
  The 
  length 
  of 
  a 
  vector 
  a 
  will 
  be 
  written 
  as 
  a 
  

   (i. 
  e. 
  with 
  the 
  bar) 
  ; 
  thus 
  ag 
  = 
  A'3, 
  unless 
  in 
  cases 
  where 
  no 
  

   ambiguity 
  can 
  arise, 
  when 
  a 
  simply 
  will 
  be 
  written 
  for 
  the 
  

   length 
  of 
  the 
  vector 
  a, 
  

  

  3. 
  If 
  we 
  agree 
  to 
  indicate 
  by 
  i^ 
  the 
  operation 
  of 
  rotating 
  

   any 
  vector 
  to 
  which 
  it 
  is 
  prefixed 
  through 
  an 
  angle 
  6 
  in 
  the 
  

   positive 
  direction, 
  then 
  

  

  i^cc 
  = 
  —a 
  or 
  ^'T 
  _ 
  — 
  2 
  

  

  and 
  

  

  TT 
  IT 
  

  

  L^a 
  = 
  (cos 
  6 
  + 
  L^ 
  sin 
  0)o(. 
  or 
  c^ 
  = 
  cos 
  6 
  -f 
  t- 
  sin 
  0. 
  

  

  Also, 
  if 
  ^ 
  = 
  Dtf, 
  to, 
  is 
  the 
  vector 
  obtained 
  by 
  increasing 
  a 
  

   in 
  length 
  D 
  times 
  and 
  then 
  rotating 
  the 
  increased 
  vector 
  

   through 
  an 
  angle 
  /in 
  the 
  positive 
  direction. 
  

  

  Plane 
  vector 
  operators 
  such 
  as 
  t 
  are 
  well 
  known 
  to 
  be 
  

   subject 
  to 
  the 
  same 
  rules 
  as 
  ordinary 
  algebraical 
  symbols. 
  

  

  