438 
ME.  A.  CAYLEY’S  EIETH  MEMOIR  EPOX  QUAXTICS. 
so  that  the  first-mentioned  relation  is  equivalent  to 
(a’  — oi) 
1, 
a -{“  a', 
ua' 
1, 
i3+i3', 
(313' 
1, 
y+y', 
yy' 
= 0, 
and  the  two  sets  give  rise  to  an  involution.  The  condition  of  homography  as  expressed 
by  the  equality  of  the  anharmonic  ratios  may  be  uTitten 
« — — a. 
« — 7.«'  — (3  — 7^a  — ’ 
or  multiplying  out, 
(a—(3)(oi—(3')(ci'  — y)(a'  — y')  — (K—^}(a'  — (3’)(a  — y)(ci'  — y')  = 0, 
which  is  a form  for  the  equation  of  involution  of  the  thi’ee  pairs.  But  this  and  the 
other  transformations  of  the  equation  of  involution  is  best  obtained  by  a difierent 
method,  as  will  be  presently  seen. 
111.  Imagine  now  any  number  of  paii’S  a,  a';  (3,  (3' ; y,  y' ; &c.  in  involution, 
and  let  x,  y,  z,  w be  the  foui'th  harmonics  of  the  same  quantit}’  X uith  respect  to  the 
pairs  «,  a! ; (3,  (3' ; y,  y'  and  h,  respectively ; then  the  anharmonic  ratios  of  the  set 
(x,  y,  z,  w)  will  be  independent  of  X,  or  what  is  the  same  thing,  if  a’',  y\  z',  lo'  ai'e  the 
fourth  harmonics  of  any  other  quantity  X'  with  respect  to  the  same  four  paii’s,  the  sets 
(.r,  y,  2,  w)  and  {x\  y\  z\  w')  will  be  homographic,  or  we  shall  have 
= 0. 
1, 
X, 
/yj 
w ^ 
xx' 
1, 
yy’ 
1, 
^ , 
zz' 
1, 
w, 
w', 
wiv' 
It  will  be  sufficient  to  show  this  in  the  case  where  X is  anything  whatever,  but  X'  has  a 
determinate  value,  say  X'=oo  ; and  since  if  all  the  terms  a,  a',  &c.  are  diminished  by  the 
same  quantity  X the  relations  of  involution  and  homography  Avill  not  be  affected,  we 
may  without  loss  of  generality  assume  X=0,  but  in  this  case 
X—  , x' ■=Uu,-\-ah 
and  the  equation  to  be  proved  is 
1, 
1, 
1, 
yV 
y+y'' 
88' 
8 + 8' 
a + a'. 
cca, , 
(3-\-(3', 
i3/3'. 
y+y', 
yy'. 
s+^'. 
= 0, 
which  is  obviously  a consequence  of  the  equations  which  express  the  involution  of  the 
four  pairs. 
