TEANSrOEMATION  OF  A BIPAETITE  QIJADEIC  FUNCTION.  43 
and  thence  
(x,  y,  z)=((tr.  Q)’-'(tr.  Q+Y)(tr.  Q — Y)-'tr.  Qjx^,  y^,  zj. 
10.  Hence,  recapitulating,  we  have  the  following  theorem  for  the  automorphic  linear 
transformation  of  the  bipartite 
(QX^,  y,  zX^,  y,  z), 
Hz.  Y being  an  arbitrary  matrix,  if 
{x,  y,  2)=(Q-'(Q-YXQ+Y)-’QX^^,  y^,  2j, 
(x,  y,  z)=((tr.  Q)-Xtr.  Q+YXtr.  Q— Y)-'tr.  Qjxp  y^,  z,), 
then 
(QJ^,  y,  2lx,  y,  z) = 3/;,  ^;Ixp  y , z ), 
which  is  the  theorem  in  question, 
11.  I have  thought  it  worth  while  to  preserve  the  foregoing  investigation,  but  the 
most  simple  demonstration  is  the  verification  a posteriori  by  the  actual  substitution  of 
the  transformed  values  of  [x,  y,  2),  (x,  y,  z).  To  effect  this,  recollecting  that  in  general 
tr,  (A“‘)=(tr.  A)“' and  tr.  ABCI)  = tr.  D.  tr.  C.  tr.  B,  tr.  A,  the  transposed  matrix  of 
substitution  for  the  further  variables  is 
Q(Q-Y)-XQ+Y)Q-*; 
and  compounding  this  with  the  matrix  Q of  the  bipartite  and  the  matrix 
Q-XQ->Y}(Q-1-Y)-'Q 
of  substitution  for  the  nearer  variables,  the  theorem  will  be  verified  if  the  result  is  equal 
to  the  matrix  Q of  the  bipartite,  that  is,  we  ought  to  have 
q(q-y)-XQ+y)q-‘qq-'(q-y)(q-i-y)-*q=q, 
or  what  is  the  same  thing, 
Q(Q-Y)-'(Q  + Y)Q-'(Q-YXQH-Y)-'Q  = Q; 
this  is  successively  reducible  to 
(Q+Y)Q-'(Q  — Y)  =(Q-Y)Q-'(Q+Y) 
Q-XQ  + Y)Q-XQ-Y)  = Q-'(Q— Y)Q-‘(Q+Y) 
(1  + Q-‘YX1  — Q-A)  =(1  — Q-'YXI  + Q-'A), 
which  is  a mere  identity,  and  the  theorem  is  thus  shown  to  be  true, 
12.  It  is  to  be  observed  that  in  the  general  theorem  the  transformations  or  matrices 
of  substitution  for  the  two  sets  of  variables  respectively  are  not  identical,  but  it  may  be 
required  that  this  shall  be  so.  Consider  first  the  case  where  the  matrix  Q is  symmetrical, 
the  necessary  condition  is  that  the  matrix  Y shall  be  skew  symmetrical ; in  fact  we  have 
tljen 
tr.  Q = Q,  tr.  Y=— Y, 
and  the  transformations  become 
(.r,  y,  z)  = (Q-'(Q-Y)(Q+Y)-'Ql4-„  z,), 
(x,  y,  z)=(ir‘((2-Y)(Q+Y)-'QXx„  y„  z,), 
G 2 
