﻿412 
  

  

  Dr. 
  T. 
  Muir 
  : 
  Aggregates 
  of 
  

  

  therefore 
  called 
  for. 
  Let 
  us 
  consider 
  the 
  case 
  of 
  2 
  

   In 
  less 
  compact 
  notation 
  it 
  is 
  

  

  1234 
  

  

  5678 
  

  

  1234 
  

  

  5678 
  

  

  1245 
  

  

  3678 
  

  

  1257 
  I 
  

   3468 
  j 
  

  

  + 
  

  

  1235 
  I 
  

  

  4678 
  I 
  

  

  1246 
  

  

  3578 
  

  

  1258 
  

   3467 
  

  

  + 
  

   + 
  

   + 
  

  

  1236 
  

  

  4578 
  

  

  1247 
  

  

  3568 
  

  

  1267 
  

   3458 
  

  

  1237 
  

  

  4568 
  

  

  1248 
  

   3567 
  

  

  1268 
  

   3457 
  

  

  + 
  

  

  + 
  

  

  1238 
  

   4567 
  

  

  1256 
  

   3478 
  

  

  | 
  1278 
  

   3456 
  

  

  + 
  

   + 
  

   + 
  

  

  1345 
  

  

  2678 
  

  

  + 
  

  

  1346 
  

  

  2578 
  

  

  — 
  

  

  1347 
  

  

  2568 
  

  

  + 
  

  

  1348 
  

  

  2567 
  

  

  — 
  

  

  1356 
  

  

  2478 
  

  

  1357 
  

  

  2468 
  

  

  - 
  

  

  1358 
  

  

  2467 
  

  

  - 
  

  

  1367 
  

  

  2458 
  

  

  , 
  j 
  1368 
  

   + 
  I 
  2457 
  

  

  - 
  

  

  1378 
  

   2456 
  

  

  1456 
  

  

  2378 
  

  

  — 
  

  

  1457 
  

  

  2368 
  

  

  + 
  

  

  1458 
  

  

  2367 
  

  

  + 
  

  

  1467 
  

  

  2358 
  

  

  

  1468 
  1 
  

   2357 
  | 
  

  

  1478 
  

   2356 
  

  

  1567 
  i 
  

  

  2348 
  j 
  

  

  | 
  1568 
  

   j 
  2347 
  

  

  1578 
  

   2346 
  

  

  C 
  

  

  + 
  

   s, 
  i. 
  <?. 
  35. 
  

  

  1678 
  

   2345 
  

  

  This 
  

  

  may 
  

  

  the 
  number 
  of 
  terms 
  being 
  of 
  cour 
  

  

  be 
  viewed 
  as 
  consisting 
  of 
  four 
  sub-aggregates, 
  A, 
  B, 
  C, 
  D, 
  

   viz. 
  A 
  occupying 
  the 
  first 
  line, 
  B 
  the 
  second 
  and 
  third 
  lines, 
  

   C. 
  the 
  fourth 
  and 
  fifth 
  lines, 
  and 
  D 
  the 
  sixth 
  and 
  seventh 
  

   lines. 
  Of 
  these 
  the 
  first 
  is 
  seen 
  to 
  be 
  a 
  Kronecker 
  aggregate. 
  

   The 
  second 
  B 
  consists 
  of 
  two 
  parts 
  which 
  cancel 
  each 
  other 
  ; 
  

  

  II 
  94 
  '3 
  I 
  

   5C8 
  ' 
  

  

  1234 
  

   diminished 
  by 
  a 
  Kronecker 
  aggregate 
  is 
  equal 
  to 
  — 
  &C7H 
  

  

  can 
  be 
  shown 
  to 
  be 
  

  

  Similarly, 
  the 
  sum 
  of 
  the 
  first 
  four 
  terms 
  of 
  C 
  

  

  cancels 
  the 
  sum 
  of 
  the 
  last 
  six 
  terms 
  : 
  and 
  the 
  sum 
  of 
  the 
  

   first 
  six 
  terms 
  of 
  D 
  cancels 
  the 
  sum 
  of 
  the 
  last 
  four 
  terms. 
  

  

  and 
  the 
  sum 
  of 
  the 
  last 
  six 
  term 
  

   1234 
  | 
  

  

  5678 
  

  

  The 
  new 
  aggregate, 
  2 
  

  

  1234 
  

  

  5678 
  

  

  , 
  is 
  thus 
  seen 
  to 
  be 
  the 
  sum 
  of 
  

  

  four 
  zeros 
  ; 
  and 
  it 
  will 
  be 
  found 
  that 
  all 
  the 
  others 
  have 
  a 
  like 
  

   constitution. 
  

  

  4. 
  The 
  four 
  sub-aggregates 
  in 
  the 
  preceding 
  are 
  easily 
  

   seen 
  to 
  be 
  

  

  1234 
  

  

  5678 
  

  

  1245 
  

  

  3678 
  

  

  1345 
  

  

  145(5 
  

  

  ^ 
  | 
  2678 
  | 
  ' 
  *\ 
  2378 
  | 
  * 
  

   The 
  last 
  three 
  are 
  not 
  essentially 
  different; 
  so 
  that 
  to 
  

  

  