﻿the 
  Alternate 
  Current 
  Generator. 
  51 
  

  

  so 
  that 
  

  

  _ 
  1 
  

  

  ^1 
  — 
  - 
  — 
  ^^ 
  ^0? 
  

  

  where 
  Si 
  is 
  the 
  mfinite 
  continued 
  fraction 
  operator 
  whose 
  

   leading 
  term 
  is 
  ^i. 
  

   Asain, 
  if 
  

  

  equations 
  VI., 
  that 
  

  

  

  

  1 
  

  

  

  

  

  

  

  

  ^2 
  = 
  

  

  = 
  "^'2 
  - 
  

  

  

  1 
  

  

  

  

  

  

  

  

  

  "^3 
  

  

  T-4 
  - 
  

  

  1 
  

  

  -&c. 
  

  

  

  

  

  83 
  = 
  

  

  ■■h- 
  

  

  1 
  

  

  1 
  

   ^5- 
  

  

  1 
  

  

  - 
  &c. 
  

  

  

  

  

  S.= 
  

  

  ■■ 
  n- 
  

  

  1 
  

  

  -&C. 
  

  

  ? 
  

  

  

  

  

  

  find 
  

  

  Viof 
  

  

  in 
  a 
  

  

  similar 
  way, 
  

  

  or 
  hj 
  

  

  making 
  

  

  use 
  

  

  of 
  

  

  nat 
  

  

  

  

  1 
  

  

  

  

  

  

  

  

  a^ 
  = 
  

  

  — 
  

  

  ^% 
  

  

  

  

  

  

  

  -'2 
  

  

  as 
  = 
  — 
  ■o-«2 
  &c., 
  &c. 
  

  

  Hence 
  for 
  the 
  complete 
  solution 
  we 
  have 
  

  

  Uq 
  = 
  — 
  S^ai 
  = 
  81^2^2 
  = 
  — 
  SA^s^-s 
  = 
  &c., 
  

  

  which 
  gives 
  aj, 
  ag, 
  ag, 
  &c., 
  u2-> 
  ^4? 
  «65 
  &(^v 
  ii^ 
  terms 
  of 
  the 
  

   known 
  vector 
  uq 
  provided 
  the 
  continued 
  fraction 
  operators 
  

   Si, 
  S25 
  S3, 
  &c., 
  are 
  determinate. 
  

  

  It 
  is 
  easily 
  seen 
  that 
  they 
  are 
  determinate 
  for 
  when 
  q 
  

   becomes 
  a 
  large 
  number 
  

  

  -^^+1 
  = 
  2- 
  = 
  Tj+3(see§5) 
  ; 
  

  

  that 
  is, 
  Si, 
  ^2, 
  S3, 
  IS4, 
  &c. 
  are 
  recurring 
  continued 
  fraction 
  

   operators, 
  and 
  the 
  recurring 
  elements 
  when 
  reached 
  are 
  

   simple 
  numbers. 
  

  

  8. 
  The 
  form 
  of 
  solution 
  obtained 
  is 
  one 
  very 
  easy 
  of 
  

   practical 
  application. 
  In 
  computing 
  the 
  continued 
  fraction 
  

   operators 
  just 
  so 
  many 
  of 
  their 
  known 
  i, 
  t 
  elements 
  (see 
  § 
  5) 
  

   need 
  be 
  taken 
  account 
  of 
  as 
  are 
  necessary 
  to 
  give 
  the 
  required 
  

   degree 
  of 
  approximation. 
  

  

  Moreover, 
  in 
  the 
  case 
  of 
  a 
  practical 
  alternator, 
  as 
  the 
  re- 
  

   sistance 
  of 
  the 
  field-magnet 
  coils 
  is 
  negligible 
  relative 
  to 
  

  

  E 
  2 
  

  

  