42 
ME.  A.  CAYLEY  ON  THE  AUTOMOEPHIC  LINEAE 
and  it  is  easy  to  see  that  the  equation  will  be  satisfied  by  writing 
( QX5  —X,  n —y,  l—z)={r  '/?,  1), 
(tr.QXS— X,  H— y,  Z— z)=  — (tr.YXS,  H,  Z), 
where  Y is  any  arbitrary  matrix.  In  fact  we  have  then 
( QX?  -y.  I -^XS,  H,  z)=  ( rjx.n.t  X^,  h,  z), 
(tr.QXH— X,  H— y,  Z— zX?,  n , 2;  ) = — (tr.YXH,  H,  ZJJ,  , yi  , 1) 
= -(  YX?,^,rX2,  H,  Z), 
and  the  sum  of  the  two  terms  consequently  vanishes. 
6.  The  equation 
(QX?-^,  n-y,  t-z)={YJX,  n,  1) 
gives 
(Q— YX?,  n, 
and  we  then  have 
(Q+YX^,  y,  zj. 
In  fact  the  two  equations  give 
(2QX^^  l)={^Jjc-\-x^,  y-\-y^,  z+zj, 
or  what  is  the  same  thing, 
2(?,  l)={x+x,,  y-\-y\,  z+z^), 
which  is  the  equation  assumed  as  the  definition  of  (^,  ti,  1) ; and  conversely,  this  equation, 
combined  with  either  of  the  two  equations,  gives  the  other  of  them. 
7.  We  have  consequently 
(x,  y,  ^)=(Q-*(Q-Y)XI,  1). 
(^,  »?,  2:)=((0+Y)-'QX^„  y,,  z,), 
and  thence 
(*,  y,  z)=(Q-{Q-Y)(Q+Y)-QX*„  y„  2,). 
8.  But  ill  like  manner  the  equation 
(tr.QXH-x,  H-y,  Z-z)= -(tr.  YXS,  H,  Z) 
gives 
and  we  then  obtain 
(tr.  Q+YX3,  H,  Z)=(tr.  Q^x,  y,  z), 
(tr.  Q-YJS,  H,  Z)=(tr.  y„  z,). 
9.  In  fact  these  equations  give 
(tr.  2QXH,  H,  Z)  = (tr.  ^X^+Xp  y+y,,  z + zj, 
or 
2(H,  H,  Z)=(x-f  Xp  y+yp  z+zj  ; 
and  conversely,  this  equation,  combined  with  either  of  the  two  equations,  gives  the  other 
of  them.  We  have  then 
(x,  y,  z)=:((tr.Q)->(tr.Q+YXH,  H,  Z), 
(H,  H,  Z)=((tr.  Q — Y)-'  tr.  QXxp  yp  zj. 
