ME.  A.  CAYLEY’S  EIETH  MEMOIE  UPON  QUAJS'TICS. 
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The  cubicovariant  may  be  obtained  by  writing  the  quartic  under  the  form 
{ax-\-hy,  bx-\-cy,  cx-\-dy^  dx-\rey\x^  yj 
and  treating  the  linear  functions  as  coefficients,  or  considering  this  as  a cubic,  the 
cubicovariant  of  the  cubic  gives  the  cubicovariant  of  the  quartic. 
If  we  represent  the  cubicovariant  by 
0=(a,  b,  c,  d,  e,  f,  gj^,  yf, 
then  we  have  identically, 
ag— 9ce4-8d^=0 ; 
and  moreover  forming  the  quadrinvariant  of  the  sextic,  we  find 
ag — 6bf+15ce — 10d^=^n, 
where  □ is  the  discriminant  of  the  quartic.  From  these  two  equations  we  find 
bf — 4ce+3d®= — 
which  is  an  expression  given  by  Mr.  Salmon  : it  is  the  more  remarkable  as  the  left- 
hand  side  is  the  quadrinvariant  of  (b,  c,  d,  e,  i\x^  yY,  which  is  not  a covariant  of  the 
quartic.  It  may  be  noticed  also  that  we  have 
af — 3be  2cd=  0 
bg — 3cf-j-2de=0. 
134.  The  CO  variants  of  the  intermediate 
aU-f  6/3H 
of  the  quartic  and  Hessian  are  as  follows,  viz. — 
The  quadrinvariant  is 
I(aU+6/3H)  = (I,  18J,  ; 
the  cubinvariant  is 
J(aU-i-6i3H)=(J,  P,  91 J,  -P-1-54P;X«>  i3)®; 
the  Hessian  is 
H(aU+6|3H)=(l,  0,  -3I^a,  (ifU 
+(0,  I,  9JI«,  ^)^U; 
and  the  cubicovariant  is 
0(«U  + 6/3H)=(I,  0,  -91,  -54J^«,  (3)^0; 
to  which  may  be  added  the  discriminant,  which  is 
□ («U+6/3H)=(l,  0,  -181,  108J,  81P,  972IJ,  -2916P^a,  i3)«a. 
135.  The  expression  for  the  lambdaic  is 
a , 5 , c—1\ 
h , c -{-X,  d 
c — 2k,  d , e 
If  the  determinant  is  represented  by  A,  that  is  if 
=J-fM-4X^ 
A=  — 4X"+AI+J, 
then  if  Xj,  A2,  X3  are  the  roots  of  the  equation  A=0,  and  if  the  values  of  B^A,  &c.  obtained 
by  writing  in  the  place  of  X are  represented  by  &c.,  then  if  x,  y satisfy  the 
3 0 2 
