434 
ME.  A.  CAYLEY’S  EIETH  IVIEMOIE  UPON  QUANTICS. 
4a  = — (a — jSy, 
2(aa')~^Q=(c(—(3)(a — j3'), 
4a'-^a'  =(a-/3')^ 
R =0 
(««')  ‘H  =((3'—j3)(a:—cc^y. 
104.  In  the  case  of  three  quadrics,  of  the  expressions  which  are  or  might  be  con- 
sidered, it  will  be  sufficient  to  mention 
(a  , b , c X^v,  yf, 
(1) 
{a! , b\  c'X^,  y)\ 
(2) 
(«",  5",  c"X^,  y)\ 
(3) 
a , b , c 
5 
(4) 
a',  5',  c' 
a",  b",  c" 
where  (1),  (2),  (3)  are  the  quadrics  themselves,  and  (4)  is  an  invariant,  linear  in  the 
coefficients  of  each  quadric.  And  where  it  is  convenient  to  do  so,  I uTite 
(1)  = u, 
(2)  = U', 
(3)  = U", 
(4)  = a. 
105.  The  equation  0 = 0 is,  it  is  clear,  the  condition  to  be  satisfied  by  the  coefficients 
of  the  three  quadrics,  in  order  that  there  may  be  a syzygetic  relation  XU-f-jU;U'-j->'U"=0. 
or  what  is  the  same  thing,  in  order  that  each  quadric  may  be  an  mtermediate  of  the 
other  two  quadrics;  or  again,  in  order  that  the  three  quadrics  maybe  in  Involution. 
Expressed  in  terms  of  the  roots,  the  relation  is 
1,  a -J-jQ  , a|3 
1,  a'j3' 
1,  a"(5" 
= 0; 
and  when  this  equation  is  satisfied,  the  three  pairs,  or  as  it  is  usually  expressed,  the  six 
quantities  a,  (3 ; a',  f3' ; a",  (5",  are  said  to  be  in  involution,  or  to  form  an  inv  olution. 
And  the  two  perfectly  arbitrary  pairs  a,  j3 ; a!,  (3'  considered  as  belonging  to  such  a 
system,  may  be  spoken  of  as  an  involution.  If  the  two  terms  of  a pair'  are  equal,  e.  g. 
if  cc"  = (3"  = d,  then  the  relation  is 
1, 
1, 
1, 
26  , 6^ 
a (3,  0.(3 
o'+(3',  o' (3' 
= 0; 
and  such  a system  is  sometimes  spoken  of  as  an  involution  of  five  terms.  Considering 
