﻿50 
  Prof. 
  T. 
  R. 
  Lyle 
  on 
  the 
  Theory 
  of 
  

  

  Note 
  that 
  the 
  vector 
  equations 
  in 
  this 
  paragraph 
  are 
  

   equations 
  connecting 
  the 
  different 
  vectors, 
  considered 
  purely 
  

   as 
  vectors, 
  without 
  any 
  reference 
  to 
  the 
  order 
  of 
  the 
  harmonic 
  

   they 
  originally 
  represented. 
  

  

  6. 
  We 
  have 
  thus 
  obtained 
  the 
  following 
  infinite 
  series 
  of 
  

   equations 
  connecting 
  the 
  vectors 
  used 
  to 
  represent 
  x 
  and 
  f 
  : 
  — 
  

  

  ^lai 
  + 
  ^2 
  

  

  = 
  — 
  «o 
  

  

  ai 
  + 
  T2a2 
  + 
  ^3 
  

  

  = 
  

  

  «2 
  + 
  ^3^3 
  -l- 
  «4 
  

  

  = 
  

  

  a3 
  + 
  T4a4-Fa5 
  

  

  = 
  

  

  (VI.) 
  

  

  &c., 
  &c. 
  

  

  And 
  as 
  it 
  is 
  well 
  known 
  that 
  algebraic 
  methods 
  are 
  applicable 
  

   to 
  plane 
  vector 
  operators 
  of 
  the 
  type 
  here 
  made 
  use 
  of, 
  we 
  

   obtain 
  the 
  following 
  infinite 
  determinant 
  vector 
  solution 
  

   for 
  ai, 
  namely, 
  

  

  ^1 
  1 
  . 
  

  

  1 
  T2 
  1 
  . 
  

  

  1 
  ^3 
  1 
  . 
  

  

  1 
  T4 
  1 
  . 
  . 
  . 
  . 
  ai 
  = 
  — 
  ^ 
  ^ 
  / 
  , 
  , 
  ^ 
  ao 
  

  

  1 
  ^5 
  1 
  

   1 
  Tg 
  

   &c., 
  &c. 
  

   or 
  

  

  Pi^i 
  = 
  — 
  Haao 
  

   where 
  Pj 
  is 
  the 
  infinite 
  determinant 
  operator 
  whose 
  leading 
  

   term 
  is 
  ^i, 
  and 
  112 
  that 
  whose 
  leading 
  term 
  is 
  Tg. 
  

  

  7. 
  Pi, 
  112, 
  P3, 
  114, 
  &c., 
  being 
  the 
  determinants 
  whose 
  

   leading 
  terras 
  are 
  fi, 
  T2, 
  ^3, 
  r^, 
  &c. 
  respectively, 
  we 
  find 
  at 
  

   once 
  by 
  expanding 
  that 
  

  

  Pi 
  = 
  iin2-P3 
  

  

  n2 
  = 
  T2P3 
  

  

  •^2 
  

  

  1 
  

  

  

  

  1 
  

  

  ^3 
  

  

  10 
  

  

  

  

  1 
  

  

  T4 
  1 
  

  

  

  

  

  

  1 
  ^5 
  1 
  ... 
  . 
  

  

  

  

  

  

  1 
  Tg 
  1 
  . 
  . 
  . 
  . 
  

  

  &c., 
  &c. 
  

  

  hence 
  

  

  p. 
  

  

  = 
  «, 
  

  

  1 
  , 
  

  

  114, 
  &c., 
  

  

  1 
  

  

  n^/Ps 
  

  

  T, 
  — 
  

  

  4-1 
  

  

  = 
  Si 
  (say), 
  

  

  T4 
  — 
  &c. 
  

  

  