ME.  A.  CAYLEY’S  FIFTH  MEMOIE  UPON  QUAHTICS. 
441 
which  is,  in  fact,  one  of  the  equations  which  express  the  equality  of  the  anharmonic 
ratios  of  (a,  j3,  y,  and  (a-,  /3',  y', 
115.  In  the  case  of  a cubic,  the  expressions  considered  are — 
(a,  b,  c,  dXx,  y)\ 
{ac—h^^  ad — be,  bd—d^X^->yTi 
L 
— a^d-\-?>abc—2b^,  ^ 
—abd-\-2ac^  — b‘^c, 
+ acd — 2b'^d  + bc^ , 
-\-ad^  — ‘S>bcd-{-2c^ 
[ y)\ 
a^d^ — Qabcd + 4ac® + 4:b^d — Sb^c^, 
(1) 
(2) 
(3) 
(4) 
where  (1)  is  the  cubic,  (2)  is  the  quadricovariant  or  Hessian,  (3)  is  the  cubicovariant,  and 
(4)  is  the  quartinvariant  or  discriminant. 
And  where  it  is  convenient  to  do  so,  I write 
so  that  we  have 
(1)  = u, 
(2)  = H, 
(3)  = O, 
(4)  = □, 
cl)2_  □u^-j-4H3=0. 
116.  The  Hessian  may  be  written  under  the  form 
(aa:+by)(cx-{-dy)  — {bx-{-cyf, 
(which,  indeed,  is  the  form  under  which  qua  Hessian  it  is  originally  given),  and  under 
the  form 
if,  —yx,  A'" 
a,  b , c 
b , c , d 
The  cubicovariant  may  be  written  under  the  form 
{2{ac  — ¥)x-\-{ad—bc)y]{bif-\-2  exy + dy^) 
— { {ad — bc)x-\-2{bd  — cfj } {ax^ + 2bxy-\-cy% 
that  is,  as  the  .Jacobian  of  the  cubic  and  Hessian ; and  under  the  form 
that  is,  as  the  evectant  of  the  discriminant. 
The  discriminant,  taken  negatively,  may  be  written  under  the  form 
+ 4(«c  — b^){bd  — &)  — {ad—bcf, 
that  is,  as  the  discriminant  of  the  Hessian. 
3 N 2 
