[ 39  ] 
III.  A Memoir  on  the  Automorphic  Linear  Transformation  of  a Bipartite  Quadric 
Function.  By  Aethue  Cayley,  Esp,  F.R.S. 
Received  December  10,  1857, — Read  January  14,  1858. 
The  question  of  the  automorphic  linear  transformation  of  the  function  t^^at 
is  the  transformation  by  linear  substitutions,  of  this  function  into  a function 
of  the  same  form,  is  in  effect  solved  by  some  formulae  of  Eulee’s  for  the  transformation 
of  coordinates,  and  it  was  by  these  formulae  that  I was  led  to  the  solution  in  the  case  of 
the  sum  of  n squares,  given  in  my  paper  “ Sur  quelques  proprietes  des  determinants 
gauches*.”  A solution  grounded  upon  an  a-priori  investigation  and  for  the  case  of  any 
quadric  function  of  n variables,  was  first  obtained  by  M.  Heemite  in  the  memoir 
“ Remarques  sur  une  Memoire  de  M.  Cayley  relatif  aux  determinants  gauchesf.”  This 
solution  is  in  my  Memoir  “ Sur  la  transformation  d’une  function  quadratique  en  elle- 
meme  par  des  substitutions  lineairesj,”  presented  under  a somewhat  different  form 
involving  the  notation  of  matrices.  I have  since  found  that  there  is  a like  transform- 
ation of  a bipartite  quadric  function,  that  is  a lineo-linear  function  of  two  distinct  sets, 
each  of  the  same  number  of  variables,  and  the  development  of  the  transformation  is  the 
subject  of  the  present  memoir. 
1.  For  convenience,  the  number  of  variables  is  in  the  analytical  formulae  taken  to  be 
3,  but  it  will  be  at  once  obvious  that  the  formulae  apply  to  any  number  of  variables  what- 
ever. Consider  the  bipartite  quadric 
Q a , b , c Jx,  y,  zjx,  y,  z) 
a!  , b' , d 
a",  b\  c" 
which  stands  for 
{ax  + % -\-cz  )x 
-\-{(ix  -\-Vy 
-\-{d'xA-y'y-\-G'z)z., 
and  in  which  {x^  y,  z)  are  said  to  be  the  nearer  variables,  and  (x,  y,  z)  the  further 
variables  of  the  bipartite. 
* Ceelle,  t.  xxxii.  pp.  119-123  (1846). 
f Cambridge  and  Dublin  Mathematical  Journal,  t.  ix.  pp.  63-67  (1854). 
+ Crelle,  t.  1.  pp.  288-299  (1855). 
