﻿54 
  Prof. 
  T. 
  R. 
  Lyle 
  on 
  the 
  Theory 
  of 
  

  

  from 
  the 
  first 
  expression, 
  or 
  

  

  = 
  -' 
  — 
  sm 
  [wt 
  + 
  bi 
  - 
  A) 
  

  

  from 
  the 
  second, 
  as 
  tx 
  = 
  Di*"-^!. 
  

  

  Similarly 
  the 
  total 
  alternating 
  e.m.f. 
  (H 
  say) 
  generated 
  

   in 
  the 
  field 
  circuit 
  is 
  given 
  by 
  either 
  

  

  TT 
  

  

  TT 
  

  

  or 
  H 
  = 
  — 
  t^ 
  Z/>T^,a^;, 
  

  

  so 
  that 
  the 
  fundamental 
  harmonic 
  of 
  H 
  is 
  equal 
  to 
  either 
  

   2(t)mrj 
  r 
  1 
  

  

  P 
  

  

  \-dn[2wt^\^Tr)-\- 
  sin(2ft)^ 
  + 
  bi 
  + 
  /32 
  + 
  b3 
  + 
  7r)l 
  

  

  2(1)1717} 
  A2 
  . 
  / 
  , 
  , 
  ^ 
  , 
  N 
  

  

  or 
  — 
  - 
  sm 
  (2aj^ 
  + 
  b^ 
  -f 
  ySg 
  - 
  ^2 
  + 
  tt). 
  

  

  11. 
  The 
  mean 
  value 
  of 
  the 
  product 
  

  

  sin 
  {aQ)t-\-6) 
  sin 
  {ba)t-\-(f>) 
  

  

  being 
  zero 
  when 
  a 
  and 
  ?> 
  are 
  unequal 
  and 
  ^ 
  cos 
  (^ 
  — 
  <^) 
  when 
  

   a 
  and 
  5 
  are 
  equal, 
  we 
  find 
  that 
  the 
  mean 
  value 
  of 
  a'^ 
  where 
  

   .T 
  = 
  l,B.g 
  in 
  

  

  and 
  the 
  mean 
  value 
  of 
  f^, 
  where 
  f 
  = 
  — 
  + 
  V^r,, 
  is 
  

   = 
  T 
  + 
  i2 
  V 
  

  

  Again, 
  for 
  the 
  same 
  reason, 
  if 
  a 
  and 
  /3 
  be 
  any 
  two 
  vectors 
  

   representing 
  harmonics 
  of 
  the 
  same 
  order 
  and 
  if 
  Say8 
  be 
  the 
  

   product 
  of 
  the 
  lengths 
  of 
  a 
  and 
  ^ 
  into 
  the 
  sine 
  of 
  the 
  angle 
  

   from 
  a 
  to 
  y8 
  measured 
  in 
  the 
  positive 
  direction, 
  then 
  the 
  mean 
  

   value 
  of 
  the 
  product 
  

  

  TT 
  

  

  t'-^a 
  into 
  yS 
  

  

  is 
  =iSa^= 
  ~iSy5«. 
  

  

  Applying 
  these 
  principles 
  to 
  the 
  determination 
  of 
  the 
  

   mean 
  value 
  Eo; 
  of 
  the 
  product 
  of 
  E 
  and 
  .r, 
  that 
  is 
  of 
  the 
  

  

  