﻿48 
  Prof. 
  T. 
  R. 
  Lyle 
  on 
  the 
  Theory 
  of 
  

  

  Again, 
  the 
  sum 
  of 
  two 
  operators 
  ajt^s 
  ^2^^^? 
  can 
  be 
  ex- 
  

   pressed 
  as 
  a 
  single 
  operator 
  Aa^, 
  say, 
  that 
  is 
  

  

  where 
  a 
  is 
  any 
  vector. 
  

  

  Using 
  the 
  expression 
  for 
  iP 
  given 
  above, 
  

  

  TT 
  

  

  A 
  (cos 
  -^-VC^ 
  sin 
  '^)ol 
  

  

  TT 
  TT 
  

  

  ai 
  (cos 
  6^ 
  + 
  (-2 
  sin 
  ^i) 
  a, 
  + 
  (22 
  (cos 
  ^2 
  + 
  ^''^ 
  sin 
  ^g)^? 
  

  

  A 
  cos 
  y^ 
  = 
  ai 
  cos 
  ^1 
  + 
  ag 
  cos 
  62 
  ; 
  

   A 
  sin 
  -^ 
  = 
  ai 
  sin 
  ^j 
  + 
  a2 
  sin 
  62. 
  

  

  A^ 
  = 
  ai^ 
  + 
  «2^ 
  + 
  2aia2 
  cos 
  (^1 
  — 
  O^), 
  

  

  and 
  , 
  , 
  <2, 
  sin 
  ^1 
  -Fao 
  sin 
  ^o 
  

  

  tan 
  -f 
  = 
  ' 
  .-. 
  

  

  ai 
  cos 
  Ui 
  + 
  cio 
  cos 
  6.^ 
  

  

  so 
  

  

  that 
  

  

  Hence 
  

  

  Again, 
  if 
  

   then 
  

   seeing 
  that 
  

  

  Up 
  = 
  ^^sin 
  (poit 
  + 
  yp), 
  

  

  J^ 
  («;») 
  = 
  i^wfp 
  sin 
  fj^cot 
  + 
  7^ 
  + 
  1 
  Y 
  

   Hence 
  for 
  x 
  and 
  f 
  as 
  expressed 
  in 
  § 
  2 
  

  

  4. 
  By 
  means 
  of 
  the 
  formula 
  

  

  2 
  sin 
  a 
  cos 
  & 
  = 
  sin 
  (a 
  + 
  ^) 
  4- 
  sin 
  (a 
  — 
  h) 
  

  

  it 
  is 
  easy 
  to 
  show 
  that 
  2^* 
  cos 
  wi, 
  w 
  here 
  a: 
  is 
  the 
  a 
  series 
  of 
  odd 
  

   order 
  vectors 
  in 
  § 
  2, 
  is 
  represented 
  by 
  the 
  series 
  of 
  even 
  order 
  

   vectors 
  of 
  which 
  the 
  one 
  of 
  the 
  pth 
  order 
  is 
  the 
  vector 
  sum 
  

   of 
  Sip_i 
  and 
  ap_j_i, 
  or 
  that 
  

  

  2^ 
  cos 
  cot 
  — 
  (ai)o 
  4- 
  (aj 
  + 
  a3)2 
  + 
  (ag 
  + 
  a5)4 
  + 
  (ag 
  + 
  2i^\ 
  -f 
  &c., 
  

  

  (ai)o 
  being 
  the 
  resolved 
  part 
  of 
  ai 
  along 
  the 
  y 
  axis, 
  that 
  is 
  

   along 
  the 
  direction 
  of 
  vectors 
  of 
  zero 
  order 
  (see 
  Note 
  a, 
  § 
  2) 
  . 
  

   Similarly 
  

  

  2f 
  cos 
  wt 
  = 
  (otQ 
  + 
  a2)i 
  + 
  («2 
  + 
  <ai4)3 
  + 
  (^4 
  + 
  ^6)5 
  + 
  &C. 
  

  

  