46 
ON  THE  TEANSFOEMATION  OE  A BIPAETITE  QUAHEIC  FUNCTION. 
vB6  — vH5  ^ 
AB~H2’  AB-H2 
— vH3  vAfl 
AB^H2  ’ AB-H2 
which,  inasmuch  as  A : B : and  C : 6 remain  arbitrary,  represents,  as  it  should  do, 
an  arbitrary  symmetrical  matrix. 
17.  Hence,  finally,  we  have  the  following  Theorem  for  the  automorphic  linear  trans- 
formation of  the  bipartite  quadric, 
y,  z), 
when  the  two  transformations  are  identical,  viz.  if  be  a skew  sjTnmetrical  matrix, 
and  if 
Y=-(i{Q-’+tr.Q-})(i{Q--tr.Q-‘})Y^+Y,; 
then  if 
(*,  y,  2)=(Q-'(Q-Y)(Q+Y)-QXa;„  y„  z,), 
(x,  y,  z)=(Q-(Q-Y)(Y +Y)-'QXx„  y„  zj; 
'S^^e  have 
(QXx,  y,  2:Xx,  y,  z)  = (QX^p  y,,  y,,  zj; 
and  in  particular. 
If  Q is  a symmetrical  matrix,  then  Y is  an  arbitrary  skew  symmetrical  matrix ; 
If  Q is  a skew  symmetrical  matrix,  then  Y is  an  arbitrary  symmetrical  matrix. 
