﻿468 
  Dr. 
  F. 
  L 
  Hitchcock 
  on 
  the 
  Operator 
  V 
  in 
  

  

  and 
  by 
  operating 
  a 
  second 
  time 
  with 
  V 
  

  

  S7 
  2 
  (r 
  k 
  t)=k(k-2)r*-yt-Mr 
  k 
  ^H 
  + 
  2kr*- 
  2 
  SpVt 
  + 
  r 
  k 
  V'^ 
  

  

  = 
  r 
  k 
  \7 
  2 
  t-k{2n 
  + 
  ki-l)r 
  k 
  - 
  2 
  t, 
  (9) 
  

  

  because 
  p 
  2 
  =— 
  r 
  2 
  , 
  and 
  $p\/t= 
  — 
  nt 
  by 
  E 
  trier's 
  theorem. 
  If 
  

   we 
  prefer 
  to 
  put 
  V 
  2 
  = 
  —A 
  w 
  © 
  ma 
  y 
  state 
  this 
  identity 
  as 
  

   Formula 
  (F) 
  

  

  &(r 
  k 
  t)=r 
  k 
  At-\-'k(2n 
  + 
  k 
  + 
  i)r 
  k 
  -% 
  . 
  . 
  (F) 
  

  

  where 
  t 
  is 
  homogeneous 
  of 
  degree 
  w. 
  

  

  If 
  £ 
  is 
  harmonic 
  we 
  may 
  conveniently 
  write 
  

  

  Formula 
  (G) 
  

  

  A(r*H„)=cr*-*H 
  B 
  , 
  . 
  . 
  . 
  , 
  , 
  (G) 
  

  

  where 
  c 
  denotes 
  the 
  constant 
  &(2ra-f& 
  + 
  1). 
  

  

  We 
  are 
  now 
  able 
  to 
  prove 
  the 
  existence 
  of 
  the 
  expansion 
  (E) 
  

   inductively, 
  by 
  showing 
  that 
  if 
  it 
  exists 
  for 
  all 
  polynomials 
  

   Fp 
  of 
  degree 
  p 
  it 
  exists 
  for 
  all 
  polynomials 
  of 
  degree 
  p 
  + 
  2. 
  

   Let 
  m=p 
  + 
  2. 
  Then 
  &F 
  m 
  is 
  of 
  degree 
  p. 
  If 
  the 
  theorem 
  

   is 
  true 
  for 
  degree 
  p 
  we 
  may 
  write 
  

  

  AF 
  m 
  =H 
  p 
  + 
  r 
  2 
  H 
  p 
  _ 
  2 
  + 
  r 
  4 
  H 
  p 
  _ 
  4 
  + 
  ... 
  j 
  . 
  . 
  .-(10) 
  

   but 
  by 
  operating 
  with 
  /\ 
  on 
  both 
  sides 
  of 
  (E) 
  

  

  AF.=cH 
  ll 
  _ 
  s 
  -fc'f 
  J 
  H;. 
  4 
  fcVB 
  ll 
  . 
  l 
  +..., 
  . 
  (11) 
  

  

  where 
  the 
  c's 
  are 
  positive 
  constants 
  by 
  formula 
  (G). 
  Since 
  

   p 
  = 
  m 
  — 
  2 
  we 
  may 
  take 
  as 
  a 
  possible 
  set 
  of 
  values 
  

  

  H 
  m 
  _ 
  2 
  = 
  ~ 
  H 
  j 
  ^m-4— 
  TT^p-2 
  ; 
  -n„ 
  ? 
  _ 
  G 
  = 
  -77 
  H 
  p 
  _ 
  4 
  , 
  &C. 
  

  

  C 
  i 
  6 
  C 
  

  

  By 
  substituting 
  these 
  values 
  in 
  (E), 
  since 
  F 
  w 
  ; 
  is 
  known, 
  H„ 
  t 
  is 
  

   known. 
  It 
  is 
  therefore 
  evident 
  that 
  the 
  expansion 
  (11) 
  is 
  

   known 
  by 
  comparison 
  with 
  (10). 
  Hence 
  (E) 
  is 
  known. 
  

   Now 
  a 
  constant 
  and 
  a 
  linear 
  expression 
  are 
  always 
  harmonic. 
  

   That 
  is, 
  (E) 
  exists 
  when 
  m 
  = 
  and 
  when 
  m=l. 
  Hence 
  it 
  

   exists 
  when 
  m 
  = 
  2 
  and 
  when 
  ?rc 
  = 
  3, 
  and 
  so 
  universally. 
  

  

  The 
  same 
  inductive 
  argument 
  shows 
  that 
  the 
  expansion 
  

   (E) 
  is 
  unique; 
  for 
  if 
  expansion 
  in 
  the 
  form 
  (10) 
  is 
  uniquely 
  

   possible 
  (11) 
  is 
  uniquely 
  possible. 
  But 
  the 
  expansion 
  is 
  

   unique 
  for 
  polynomials 
  of 
  degree 
  and 
  1, 
  hence 
  for 
  all 
  

   polynomials. 
  

  

  