TEAJiSFOEMATION  OP  A BIPAETITE  QUADEIC  FUNCTION. 
41 
of  the  substitutions, 
a , b , c 
Jx,  y,  ^Xx,  y,  z)=(( 
1 , m , n ] 
i a , b , c 1 
1 , m , n 
\ 
a',  b\  d 
1',  m',  n' 
b\  d 
1' , m' , n' 
a!\  b\  c" 
1",  m",  n" 
d\  b",  d' 
r,  m",  n" 
that  is,  the  matrix  of  the  transformed  bipartite  is  obtained  by  compounding  in  order  first 
or  furthest  the  transposed  matrix  of  substitution  of  the  further  variables,  next  the  matrix 
of  the  bipartite,  and  last  or  nearest  the  matrix  of  substitution  of  the  nearer  variables. 
4.  Suppose  now  that  it  is  required  to  find  the  automorphic  linear  transformation  of 
the  bipartite 
( « , 5 , c Jx,  y,  2:Xx,  y,  z), 
a! , b' , d 
c" 
or  as  it  will  henceforward  for  shortness  be  written, 
{Qjx,  y,  ^Xx,  y,  z). 
This  may  be  effected  by  a method  precisely  similar  to  that  employed  by  M.  Heemite  for 
an  ordinary  quadric.  For  this  purpose  write 
, y+y^=2n  , z+z,=2l, 
x+x^=2H,  y+y^=2H,  z+z^=2Z, 
or  as  these  equations  may  be  represented, 
y+y,,  z+z,)=2(?,  , r), 
(x+x„  y+y^,  z+zj=2(H,  H,  Z), 
we  ought  to  have 
{QX21—X,  2ri—y,  22:— zX^S— X,  2H— y,  2Z— z)=(QX^,  y,  zJil,  y,  z). 
5.  The  left-hand  side  is 
4(QIS,  ».  CIS.  H,  Z)-2(nl3:,  y,  213,  H,  Z)-2(QIf,  ,,  IJx,  y,  z)+{njx,  y,  2lx,  y,  z), 
and  the  equation  becomes 
2(QI5,  ,,  IXS,  H,  Z)-(QXs,  y,  213,  H,  Z)-(CiI5,  ,,  tjx,  y,  z)=0, 
or  as  it  may  be  written, 
(nif.  1,  CIS,  H,  Z)-(QIx,  y,  213,  H,  Z)  I ^ 
+(QIf,  >1,  CI3,  H,  Z)— (Qlf,  1),  IJx,  y,  z)  J 
or  again,  ^ 
{QJi-s,  n-y,  l-z'l'S,  H,  Z)  I 
+ (QIC,  VI,  ZI3— X,  H — y,  Z— z)  J 
or  what  is  the  same  thing, 
{QXi-x,  ,-y,  C-zlH,  H,  Z) 
+ (tr,QI3-x,  H-y,  Z-zI?,  ,,  1)  J “ ’ 
MDCCCLVin. 
G 
