﻿Minors 
  of 
  an 
  Axkymmetric 
  Determinant. 
  411 
  

  

  which 
  according 
  to 
  Kronecker 
  vanishes 
  when 
  the 
  parent 
  

  

  determinant 
  

  

  12345678 
  | 
  • 
  

  

  1 
  oq 
  i'-r7q 
  * 
  s 
  axisymmetric, 
  may 
  be 
  written 
  

  

  2 
  I 
  1234 
  

  

  ^ 
  ' 
  5678 
  

  

  it 
  being 
  understood 
  that 
  in 
  each 
  term 
  under 
  the 
  sign 
  o£ 
  sum- 
  

   mation 
  the 
  row-numbers 
  are 
  taken 
  in 
  natural 
  order 
  and 
  the 
  

   column-numbers 
  in 
  order, 
  and 
  that 
  the 
  sign-factor 
  of 
  the 
  

   term 
  is 
  ( 
  — 
  l) 
  u 
  where 
  v 
  is 
  the 
  number 
  of 
  inverted-pairs 
  in 
  the 
  

   single 
  line 
  of 
  numbers 
  formed 
  by 
  writing 
  the 
  column-numbers 
  

   immediately 
  after 
  the 
  row-numbers. 
  

  

  2. 
  Kronecker's 
  theorem 
  was 
  originally 
  published 
  in 
  1882, 
  

   and 
  since 
  that 
  date 
  considerable 
  additional 
  light 
  has 
  been 
  

   thrown 
  on 
  it, 
  including 
  such 
  light 
  as 
  comes 
  from 
  generalization. 
  

   The 
  first 
  generalization 
  was 
  pointed 
  out 
  in 
  1897 
  * 
  and 
  the 
  

   second 
  in 
  1901 
  f« 
  The 
  latter, 
  due 
  to 
  Professor 
  Metzler, 
  is 
  

   noteworthy 
  because 
  of 
  its 
  width 
  ; 
  and 
  on 
  this 
  and 
  other 
  

   grounds 
  deserves 
  some 
  special 
  attention 
  such 
  as 
  the 
  present 
  

   paper 
  aims 
  at 
  giving 
  to 
  it. 
  In 
  effect, 
  Professor 
  Metzler 
  

  

  that 
  not 
  only 
  2 
  j 
  ^34 
  

  

  5678 
  

  

  says 
  

  

  anishos 
  in 
  the 
  circumstances 
  

  

  referred 
  to, 
  but 
  also 
  2 
  

  

  1234 
  

  

  5678 
  

  

  and 
  2 
  

  

  1234 
  

  

  5678 
  

  

  in 
  other 
  

  

  words, 
  that 
  Kronecker 
  need 
  not 
  have 
  confined 
  the 
  propo- 
  

   sition 
  to 
  the 
  case 
  where 
  n— 
  1 
  of 
  the 
  row-numbers 
  are 
  

   invariable, 
  because 
  all 
  that 
  is 
  necessary 
  is 
  that 
  the 
  number 
  

   of 
  such 
  invariable 
  row-numbers 
  shall 
  be 
  greater 
  than 
  

   and 
  less 
  than 
  n. 
  The 
  mode 
  of 
  investigation 
  is 
  that 
  em- 
  

   ployed 
  in 
  my 
  second 
  proof 
  of 
  Kronecker's 
  theorem, 
  viz., 
  

   expansion 
  of 
  each 
  member 
  of 
  the 
  aggregate 
  in 
  terms 
  of 
  

   products 
  of 
  complementary 
  minors 
  and 
  re-condensation 
  re- 
  

   sulting 
  from 
  the 
  union 
  of 
  such 
  of 
  these 
  products 
  as 
  have 
  a 
  

   factor 
  in 
  common. 
  

  

  3. 
  The 
  main 
  point 
  calling 
  for 
  notice 
  in 
  the 
  new 
  generali- 
  

   zation 
  is 
  the 
  manner 
  in 
  which 
  the 
  vanishing 
  of 
  the 
  aggregates 
  

   comes 
  about. 
  A 
  slight 
  examination 
  of 
  a 
  special 
  case 
  makes 
  

   clear 
  that 
  in 
  this 
  respect 
  there 
  is 
  an 
  essential 
  difference 
  

   •between 
  the 
  new 
  and 
  the 
  old, 
  and 
  that 
  further 
  investigation 
  is 
  

  

  * 
  Transactions 
  ~Roy. 
  Soc. 
  Edinburgh, 
  vol. 
  xxxix. 
  p. 
  226. 
  

   -f 
  Transactions 
  American 
  Math. 
  Soc. 
  vol. 
  ii. 
  pp. 
  395-403. 
  

  

  2E2 
  

  

  