ME.  A.  CAYLEY’S  FIETH  MEMOIE  UPON  QUANTICS. 
433 
100.  When  the  quadrics  are  expressed  in  terms  of  the  roots,  we  have 
={x—ay){x—^y), 
^{x — o!!y)[x — ^'y), 
4«-^n  =_(«— j3)", 
2(aa')-^Q=2cc(3 + 2«'/3'  - («  +|8)(a' +i3'), 
ia'-^n'  =-(ci'-j3J, 
{aa'y^R  ={a-  a'){oi  - |3')(i3  - «')0  - /3'), 
y\  2yx  , x'^ 
1 , a + |3,  a/3 
1 , a'+|3',  a'/3' 
101.  The  comparison  of  the  last-mentioned  value  of  E,  with  the  expression  in  terms 
of  the  roots  obtained  from  the  equation 
— E=4d  — Q^ 
gives  the  identical  equation 
(a_^)^(«'_|3')^_{2«/3+2«'/3'-(«+/3X«'+/3')}^=-4(a-a>-/3')(/3-a')(/3-/3'), 
which  may  be  easily  verified. 
102.  We  have  identically 
2a|3  + 2a'(3'-(a+/3X«'+/3') 
= 2(a -a')(cc-(3')-(a-f3)(2ci-a'-f3') 
= 2(f3-a')((3-f3')-((3  -aX2/3  -«'-/3') 
= 2(a'  — ci)(c('  — 18)  — (a' — j3')(2a'  — a — (3) 
= 2(l3'-a)(0'-(3)-(f3'-u'X2f3'-a-l3); 
and  the  equation  Q=ac'  — 2dd'-{-ca'  = 0 may  consequently  be  written  in  the  several 
forms 
« — «' 
2 
1 
1 1 
1 
R 
1 
va 
1 
R_ 
2 
1 
, 1 , 
«'  — « 
2 
1 J 
L 1 , 
so  that  the  roots  (a,  (3),  (cc',  (B')  are  harmonically  related  to  each  other,  and  hence  the 
notion  of  the  harmonic  relation  of  the  two  quadrics. 
103.  In  the  case  where  the  two  quadrics  have  a common  root  a=a'. 
a-‘U  =(x—ay)(x—f3y), 
a'~^lJ'=(x—ay)(x—(3'y), 
3 M 2 
