﻿HAUGHTON 
  — 
  GRANITES 
  OF 
  IRELAND. 
  271 
  

  

  are 
  found, 
  the 
  remaining 
  equations 
  may 
  be 
  used 
  to 
  check 
  and 
  test 
  

   the 
  original 
  hypothesis 
  in 
  a 
  manner 
  which 
  shall 
  be 
  fully 
  explained 
  

   in 
  the 
  course 
  of 
  the 
  discussion. 
  

  

  Let 
  u, 
  x, 
  y, 
  z 
  denote 
  the 
  per-centage 
  composition 
  of 
  the 
  rock 
  in 
  

   terms 
  of 
  the 
  four 
  given 
  minerals, 
  of 
  which 
  one 
  (u) 
  is 
  supposed 
  to 
  be 
  

   quartz. 
  Let 
  the 
  rock 
  be 
  analyzed, 
  and 
  let 
  A 
  v 
  B, 
  C 
  denote 
  the 
  per- 
  

   centage 
  of 
  oxygen 
  present 
  in 
  the 
  rock 
  as 
  silica, 
  as 
  peroxide, 
  and 
  as 
  

   protoxide 
  ; 
  and 
  let 
  a 
  v 
  b, 
  c 
  denote 
  the 
  per-centage 
  of 
  oxygen 
  present, 
  

   as 
  silica, 
  peroxide, 
  and 
  protoxide, 
  in 
  the 
  mineral 
  x 
  ; 
  a^, 
  b', 
  c', 
  that 
  

   in 
  the 
  mineral 
  y 
  ; 
  and 
  a", 
  b" 
  ', 
  c", 
  that 
  in 
  the 
  mineral 
  z 
  ; 
  and 
  let 
  q 
  

   denote 
  the 
  per-centage 
  of 
  oxygen 
  in 
  quartz. 
  We 
  find 
  immediately 
  

   the 
  following 
  equations 
  : 
  — 
  

  

  100 
  A 
  l 
  =qu-\-a 
  l 
  x-\-a 
  l 
  'y+a 
  1 
  "z} 
  

  

  100 
  B 
  = 
  bx 
  +b'y 
  +b"z 
  \ 
  (1) 
  

  

  100 
  C 
  = 
  ex 
  +c'y 
  +c"z 
  J 
  

  

  100=u+x+y-{-z. 
  

  

  Eliminating 
  u 
  from 
  these 
  equations, 
  and 
  replacing 
  A 
  l 
  — 
  q 
  by 
  A, 
  

   a 
  x 
  — 
  q 
  by 
  a, 
  a^^-q 
  by 
  a', 
  and 
  a"—q 
  by 
  a", 
  we 
  obtain 
  

  

  100 
  A=ax+a'y+a"z) 
  

  

  100 
  B=bx+b'y 
  + 
  b"z 
  V 
  (2) 
  

  

  lQ0C=cx+c'y 
  + 
  c"z) 
  

  

  The 
  granite 
  of 
  Donegal 
  is 
  composed, 
  as 
  I 
  have 
  already 
  shown, 
  of 
  

   four 
  minerals, 
  namely, 
  Quartz, 
  Orthoclase, 
  Oligoclase, 
  and 
  Black 
  

   Mica. 
  In 
  order 
  to 
  make 
  use 
  of 
  the 
  preceding 
  equations, 
  we 
  must 
  fix 
  

   precisely 
  the 
  values 
  of 
  a, 
  b, 
  c 
  ; 
  a', 
  V 
  , 
  d 
  ; 
  a", 
  b", 
  c", 
  by 
  means 
  of 
  the 
  

   chemical 
  composition 
  of 
  the 
  constituent 
  minerals, 
  while 
  the 
  values 
  of 
  

   A, 
  B, 
  C 
  depend 
  on 
  the 
  analysis 
  of 
  each 
  individual 
  granite 
  -specimen. 
  

   This 
  is 
  done 
  by 
  means 
  of 
  Table 
  III., 
  from 
  which 
  we 
  find 
  the 
  fol- 
  

   lowing 
  : 
  — 
  

  

  2=51-92 
  a 
  x 
  =32-81 
  

   < 
  =31-11 
  

  

  ^"=18-78 
  

  

  b 
  = 
  8-91 
  

   b' 
  =11-41 
  

   c" 
  =16-28 
  

  

  c 
  =3-55 
  

   c' 
  =3-61 
  

   c"=3-94 
  

  

  rom 
  which 
  we 
  obtain 
  also 
  

  

  a 
  BSB- 
  

  

  a'=- 
  

  

  a" 
  = 
  - 
  

  

  -19-11 
  

  

  -20-81 
  

   -33-14 
  

  

  

  and, 
  finally, 
  equations 
  (2) 
  become, 
  on 
  banishing 
  the 
  decimals, 
  

  

  519200- 
  10000A 
  1 
  =191L*?+2081 
  2 
  / 
  + 
  3314,n 
  

  

  10000B 
  = 
  891a?+1141y 
  + 
  16282 
  > 
  .... 
  (3) 
  

   10000C 
  = 
  355a?+ 
  36±2/+ 
  394z 
  J 
  

  

  The 
  solution 
  of 
  equations 
  (2) 
  or 
  (3) 
  is 
  effected 
  by 
  finding 
  nine 
  

   multipliers, 
  thus 
  defined 
  : 
  — 
  

  

  =b'c" 
  

  

  -6V 
  

  

  fi 
  =e'a" 
  -e"a' 
  

  

  y 
  =a'b" 
  —a"b' 
  

  

  ' 
  =b"c 
  

  

  -be" 
  

  

  /3' 
  =c"a 
  —ea" 
  

  

  y 
  =a"b 
  —ab" 
  

  

  "=&' 
  

  

  -b'c 
  

  

  P"=zca' 
  —e'a 
  

  

  y 
  "z=aV 
  -a'b 
  

  

  