﻿the 
  Alternate 
  Current 
  Generator. 
  49 
  

  

  Again, 
  by 
  means 
  of 
  the 
  formula 
  

  

  2 
  sin 
  a 
  sin 
  5 
  = 
  cos 
  (a 
  — 
  b') 
  — 
  cos 
  (a 
  + 
  5), 
  

   it 
  is 
  easy 
  to 
  show 
  that 
  

  

  TT 
  TT 
  TT 
  

  

  where 
  q 
  is 
  odd 
  with 
  a 
  similar 
  result 
  for 
  the 
  product 
  2x 
  sin 
  g)^. 
  

   5. 
  If 
  we 
  now 
  substitute 
  the 
  vector 
  expressions 
  from 
  

   § 
  § 
  2, 
  3, 
  4, 
  in 
  equations 
  I. 
  and 
  equate 
  separately 
  to 
  zero 
  each 
  

   set 
  of 
  vector 
  terms 
  of 
  the 
  same 
  order 
  we 
  obtain 
  the 
  two 
  series 
  

   of 
  vector 
  equations 
  

  

  ra^ 
  + 
  g6,i2|za^ 
  + 
  |(«^.l 
  + 
  a^^l)|= 
  0, 
  (IV.) 
  

  

  pci.p+pcoa^xap+ 
  ji&jj^i 
  + 
  a^+i) 
  I 
  = 
  0, 
  (V.) 
  

  

  together 
  with 
  f 
  q 
  = 
  ^vIp^ 
  where 
  q 
  is 
  any 
  odd 
  number 
  and 
  p 
  

   any 
  even 
  number. 
  

  

  From 
  TV. 
  we 
  deduce 
  a 
  series 
  of 
  equations 
  of 
  the 
  type 
  

  

  TT 
  

  

  or 
  

  

  where 
  t^ 
  is 
  the 
  operator 
  D^t""-^^, 
  in 
  which 
  

  

  / 
  2r 
  

  

  Dn 
  cos/^ 
  = 
  2 
  - 
  , 
  D<y 
  sin/^ 
  = 
  -^^— 
  , 
  

  

  that 
  is 
  

  

  Similarly 
  from 
  V. 
  we 
  deduce 
  the 
  series 
  

  

  where 
  t^ 
  is 
  the 
  operator 
  ApC~^p, 
  in 
  which 
  

  

  A^, 
  cos 
  6^ 
  = 
  2- 
  , 
  Ao 
  sin 
  ^j, 
  = 
  — 
  ^, 
  

   or 
  

  

  P/i/Z. 
  ilfa^. 
  S. 
  6. 
  Vol. 
  18. 
  No. 
  103. 
  Ji^Zz/ 
  1909. 
  E 
  

  

  