﻿52 
  Prof. 
  T. 
  R. 
  Lyle 
  on 
  the 
  Theory 
  of 
  

  

  their 
  reactance, 
  all 
  the 
  r 
  elements 
  are 
  practically 
  simple 
  

   numbers 
  independent 
  o£ 
  the 
  field 
  resistance, 
  while 
  for 
  the 
  t 
  

   elements 
  q 
  is 
  never 
  a 
  large 
  number 
  when 
  tqj^2 
  differs 
  little 
  

   from 
  tq. 
  So 
  that 
  we 
  could 
  obtain 
  S^ 
  with 
  considerable 
  accu- 
  

   racy 
  by 
  assuming 
  the 
  recurring 
  stage 
  to 
  be 
  reached, 
  and 
  

   therefore 
  

  

  ^q 
  = 
  tq 
  1 
  

  

  which 
  gives 
  the 
  quadratic 
  in 
  S^ 
  

  

  from 
  which 
  S^ 
  can 
  be 
  obtained 
  by 
  ordinary 
  algebra. 
  

  

  [In 
  solving 
  this 
  quadratic 
  the 
  two 
  operators 
  that 
  come 
  

   under 
  the 
  square-root 
  symbol 
  will 
  have 
  to 
  be 
  reduced 
  by 
  the 
  

   addition 
  theorem 
  in 
  § 
  3 
  to 
  a 
  single 
  operator, 
  at^ 
  say, 
  and 
  the 
  

   square 
  root 
  of 
  this 
  is 
  Va^^J] 
  

  

  If 
  S^ 
  obtained 
  in 
  either 
  of 
  these 
  ways 
  be 
  = 
  s^r^^^ 
  then 
  as 
  

  

  

  1 
  

  

  Sg_i 
  can 
  be 
  obtained 
  by 
  the 
  addition 
  theorem, 
  and 
  so 
  on 
  

   for 
  S^_2, 
  Sg_3, 
  &c., 
  up 
  to 
  Si. 
  Let 
  the 
  results 
  be 
  written 
  

  

  Si 
  = 
  Sit~^\ 
  ^2 
  = 
  0"2t'~^% 
  S3 
  = 
  53t~^^, 
  &C., 
  

  

  and 
  in 
  general 
  

  

  S^ 
  = 
  S^r^q, 
  tp 
  = 
  O-pL 
  ^P. 
  

  

  9. 
  As 
  cxq 
  is 
  the 
  vector 
  of 
  length 
  2i]/p 
  lying 
  along 
  the 
  axis 
  

   of 
  y 
  (phase 
  =7r/2), 
  and 
  as 
  

  

  ai 
  = 
  -g^ 
  Wi 
  = 
  - 
  -i^<«o)i 
  

  

  Again, 
  as 
  

  

  «2 
  = 
  - 
  y- 
  (ai)2 
  = 
  - 
  ^- 
  A^<ai)2, 
  

  

  __ 
  2v 
  

  

  Ci2 
  = 
  

  

  Si(T2p 
  

  

  sin 
  [2cot+~ 
  +^1 
  + 
  ^52^ 
  

  

  