﻿Minors 
  of 
  an 
  Axisymmetric 
  Deter 
  miiia 
  

  

  nt, 
  

  

  413 
  

  

  Kronecker's 
  fundamental 
  aggregate 
  % 
  

  

  1234 
  

  

  5678 
  

  

  there 
  lias 
  to 
  

  

  be 
  added 
  one 
  other, 
  2 
  

  

  1245 
  

  

  3678 
  

  

  , 
  the 
  common 
  characteristic 
  of 
  

  

  the 
  two 
  being 
  the 
  invariability 
  of 
  the 
  line-numbers 
  1, 
  2, 
  3. 
  

  

  1245 
  

  

  The 
  vanishing 
  of 
  this 
  new 
  aggregate, 
  2 
  

  

  3678 
  

  

  is 
  

  

  established 
  by 
  separating 
  the 
  four 
  terms 
  of 
  it 
  which 
  have 
  

   4 
  as 
  a 
  row-number 
  from 
  the 
  six 
  terms 
  which 
  have 
  4 
  as 
  a 
  

   column-number, 
  and 
  showing 
  that 
  the 
  one 
  group 
  cancels 
  the 
  

   other. 
  Thus 
  

  

  1245 
  

   3678 
  

  

  = 
  2 
  

  

  1245 
  

  

  3678 
  

  

  + 
  2 
  

  

  1256 
  

   3478 
  

  

  1243 
  

  

  5678 
  

  

  , 
  I 
  1234 
  

  

  + 
  5678 
  

  

  The 
  penultimate 
  step 
  here 
  presents 
  two 
  expressions 
  for 
  

  

  1234 
  

   5678 
  

  

  The 
  second 
  of 
  these 
  is 
  new 
  ; 
  the 
  firstjs 
  not, 
  since 
  

  

  1234 
  

  

  the 
  vanishing 
  of 
  Kronecker's 
  aggregate 
  2 
  

   be 
  viewed 
  as 
  a 
  statement 
  that 
  

  

  5678 
  ma 
  7 
  also 
  

  

  1234 
  

  

  5678 
  

  

  1235 
  

   4678 
  

  

  where, 
  as 
  is 
  often 
  convenient 
  in 
  dealing 
  with 
  sub-aggregates, 
  

   the 
  typical 
  term 
  on 
  the 
  right 
  is 
  reckoned 
  positive 
  and 
  the 
  

   others 
  determined 
  from 
  it. 
  We 
  are 
  thus 
  led 
  to 
  a 
  series 
  of 
  

   identities 
  

  

  1234 
  

   5678 
  

  

  _ 
  ^ 
  1 
  1235 
  I 
  _ 
  ^ 
  

  

  ~^ 
  | 
  4678 
  | 
  ^ 
  

  

  1256 
  I 
  _ 
  ^ 
  

   3478 
  - 
  ^ 
  

  

  1567 
  

   2348 
  

  

  5. 
  If 
  in 
  the 
  case 
  of 
  five- 
  line 
  minors 
  we 
  denote 
  the 
  funda- 
  

   mental 
  Kronecker 
  aggregate, 
  

  

  

  2l 
  12345 
  l 
  bvK- 
  

   Z 
  1 
  6789r 
  1 
  D/ 
  iV 
  ' 
  

  

  

  zler 
  aggregates 
  

  

  

  12345 
  1 
  ^ 
  

  

  6789 
  r|' 
  

  

  12345 
  1 
  ^ 
  ! 
  12345 
  

  

  6789 
  7-1' 
  Z 
  |6789r 
  

  

  , 
  by 
  M 
  l3 
  M 
  2 
  , 
  M 
  3 
  ; 
  

  

  