40 
ME.  A.  CAYLEY  ON  THE  AUTOMOEPHIC  LINEAE 
2.  It  is  clear  that  we  have 
b\  d 
b,  b\  b" 
a\  b\  d' 
c,  d,  d' 
and  the  new  form  on  the  right-hand  side  of  the  equation  may  also  be  written 
(tr.(  a , h , c ) Xx,  y,  y,  z), 
I , V , d 
I V\  d' 
that  is,  the  two  sets  of  variables  may  be  interchanged  provided  that  the  matrix  is  trans- 
posed. 
3.  Each  set  of  variables  may  be  linearly  transformed : suppose  that  the  substitutions 
are 
(x,  y,  z)=(  I , m , n Vn 
I',  m',  n' 
I",  m",  n" 
and 
(x,  y,  z)  = r 1 , r , 1"  X^r  Jr  zJ 
m,  m',  m" 
n , n' , n" 
Then  first  substituting  for  (x,  y,  z)  their  values  in  terms  of  [x,,  y^,  zj,  the  bipartite 
becomes 
((  a , 6 , c Vr  y.  z) 
a',  A',  d 
1' , m',  n' 
a",  d' 
Z",  m”,  w" 
Represent  for  a moment  this  expression  by 
( A , B , c , x^/.  y,^  2;Xx,  y,  z), 
A',  B',  C, 
A",  B",  C", 
then  substituting  for  (x,  y,  z)  their  values  in  terms  of  (x^  y,,  z^  it  is  easy  to  see  that  the 
expression  becomes 
((l,m,n  )X^p  y,.  2yXX;,  E,  zj, 
1',  m',  n' 
A',  B',  C 
1",  m",  n" 
A",  B",  C" 
and  re-establishing  the  value  of  the  auxiliary  matrix,  we  obtained,  finally,  as  the  result 
