44 
ME.  A.  CATLET  ON  THE  AIJTOMOEPHIC  LINEAE 
which  are  identical.  We  may  in  this  case  suppose  that  the  two  sets  of  variables 
become  equal,  and  we  have  then  the  theorem  for  the  automorphic  linear  transformation 
of  the  ordinary  quadric 
(QX^,  y,  z)\ 
viz.  Y being  a skew  symmetrical  matrix,  if 
{x,  y,  z)  = {Q  ‘(Q-Y-)(Q+Y)  'QX^p  y„  zj, 
then 
(QX^,  y,  z)^=(QX^p  Vp  ^,)'- 
13.  Next,  if  the  matrix  Q be  skew  symmetrical,  the  condition  is  that  the  matrix  Y 
shall  be  symmetrical;  we  have  in  this  case  tr.  Q=  — Q,  tr.  Y = Y,  and  the  four  factors  in 
the  matrix  of  substitution  for  (x,  y,  z)  are  respectively  —Q~\  — (Q  — Y),  — (Q-j-Y)~* 
and  — Q,  and  such  matrix  of  substitution  becomes  therefore,  as  before,  identical  •v\ith 
that  for  (x,  y,  z) ; we  have  therefore  the  following  theorem  for  the  automoi-phic  linear 
transformation  of  a skew  symmetrical  bipartite 
(QX-^’  y’  Y’  z)’ 
when  the  transformations  for  the  two  sets  of  variables  are  identical,  \iz.  Y being  any 
symmetrical  matrix,  if 
(x,  y,  2)  = (Q-‘(Q-Y)(Q+Y)-'QX^p  yp  zj, 
then 
(x,  y,  z)=(Q-*(Q— Y)(Q  + Y)  'QX^p  yp  z,), 
(QX^,  y,  zXx,  y,  z)=(QX^p  yp  ^ylxp  yp  zj- 
14.  Lastly,  in  the  general  case  where  the  matrix  Q is  anything  whatever,  the  con- 
dition is 
Q-'Y  = -(tr.  Q)-'tr.  Y 
for  assuming  this  equation,  then  first 
Q-'(Q— Y)=:(tr.  Q)-*(tr.  Q-f  Y), 
and  in  like  manner 
Q-‘(Q-1-Y)  = (tr.  Q)-'(tr.  Q — Y). 
But  we  have 
l=(tr.  Q)-*(tr.  Q— YXtr.  Q-Y)-*tr.  Q, 
and  therefore,  secondly, 
(Q-fY)-'Q  = (tr.  Q— Y)-‘tr.  Q ; 
and  thence 
q-‘(Q— YXO  + Y)-‘Q  = (tr.  Q)-Xtr.  Q+YXtr.  Q— Y)-Hr.  Q, 
or  the  two  transformations  are  identical. 
15.  To  further  develope  this  result,  let  be  expressed  as  the  sum  of  a symmetrical 
matrix  and  a skew  symmetrical  matrix  Q,,  and  let  Y be  expressed  in  like  manner  as 
the  sum  of  a symmetrical  matrix  Y^,  and  a skew  symmetrical  matrix  Y,.  We  have  then 
