ME.  A.  CAYLEY’S  FIFTH  IMEMOIE  UPON  QUANTICS. 
439 
A set  homographic  with  x,  z,  w,  which  are  the  fourth  harmonics  of  any  quantity 
whatever  X with  respect  to  the  pau’s  in  involution,  ot,  cd ; /3,  /3' ; y,  y' ; S,  is  said  to  be 
homographic  with  the  four  pair's,  and  we  have  thus  the  notion  of  a set  of  single  quan- 
tities homographic  with  a set  of  pau’s  in  involution.  This  very  important  theory  is  due 
to  M.  Chasles. 
112.  Let  r;  s;  if,  be  the  anharmonic  ratios  of  a set  a,  j3,  y,  S,  and  let  ; 5, ; be  the 
anhai’monic  ratios  (corresponding  or  not  corresponding)  of  a set  j3p  yp  And 
suppose  that /;  s' ; t' ; r\;  s\;  t'y,  r" ; s" ; f;  s" ; f;  r"' ; s'";  f;  f, 
are  the  analogous  quantities  for  thi'ee  other  pairs  of  sets ; then  an  equation  such  as 
= 0, 
1, 
n-y 
ss, 
or  as  it  is  more  conveniently  written. 
, 
rs  , 
rs  , 
s's'  , 
r's'  , 
r's  , 
rp  ry* 
n n 
s s , 
//  // 
rs  , 
//  // 
rs  , 
H H 
y*  y* 
/ 
///  m 
s s , 
nt  /// 
r 5 , 
r's"^ 
///  n 
y*  y* 
/ 
is  a relation  independent  of  the  particular  ratios  r : s which  have  been  chosen  for  the 
anharmonic  ratios  of  the  sets ; this  is  easily  shown  by  means  of  the  equations 
r+5+f=0,  ry+Sy-h?fy=0, 
which  connect  the  anharmonic  ratios.  The  equation  in  fact  expresses  a certain  relation 
between  four  sets  (a,  jS,  y,  and  four  other  sets  (osy,  (3^,  y^,  h^);  a relation  which  may 
be  termed  the  relation  of  the  homography  of  the  anharmonic  ratios  of  four  and  four 
sets : the  notion  of  this  relation  is  also  due  to  M.  Chasles. 
113.  The  general  relation 
= 0 
1, 
a +/3  , 
aj8 
1, 
«'  +/3', 
a'(3' 
1, 
may  be  exhibited  in  a great  variety  of  forms.  In  fact,  if  the  determinant  is  denoted  by 
T,  then  multiplying  by  this  determinant  the  two  sides  of  the  identical  equation 
—u,  1 ={u—v){v—w){w—u), 
we  obtain 
, —V,  1 
—w,  1 
T{ii—v){v—w)(w—u)= 
(u—oi  )(u—(3  ), 
(u—a'  )(u—f3' ), 
3 N 
{v—K  ){v—(3  ), 
(v—a')(v—(3'), 
(v—a'')(v—(3''), 
(w  — cc  ){w—(3  ) 
(w  — 06’)(w—(3') 
{tV  — od'){w—(3") 
MDCCCLVIII. 
