442 
ME.  A.  CAYLEY’S  FIETH  IVIEMOIE  YPOY  QrA^'TICS. 
117.  We  have 
(«,  c,  dyhcif-\-2cxy-\-dy^,  —a(f—2bxy—cy^f='[I^, 
which  expresses  that  a transmutant  of  the  cubic  is  the  product  of  the  cubic  and  the 
cubicovariant.  The  equation 
{(da,  di,  da,  dJJj,  —xyyn=2\J^ 
expresses  that  the  second  evectant  of  the  discriminant  is  the  square  of  the  cubic. 
The  equation 
~Scd  , 
— 2>bd-{-^c^  , 
— 35c  — \2ad, 
— 2>cd  , 
— 3c^+125c?, 
— ‘^ad—^bc , 
— 3ac+65^  , 
— 35c?+6c^  , 
— 2>ad—^bc , 
— 35^+12(?c, 
3«5  , 
— 35c+2«fZ  =27  □- 
— 3«c+65^ 
— 3«5 
cd- 
expresses  that  the  determinant  formed  with  the  second  differential  coefficients  of  the 
discriminant  gives  the  square  of  the  discriminant. 
The  covariants  of  the  intermediate  aU+|80  are  as  follows,  \'iz. — 
118.  For  the  Hessian,  we  have 
for  the  cubicovariant. 
H(«U+i30)  = (l,  0,  - n%a,  /3fH 
O(aU+/3O)=(0,  □,  0,  - %a,  /3}^U 
+(1,  0,  05:«,/3)^O 
= (a^-i3^n)(acD+^DU); 
and  for  the  discriminant. 
□ (aU+j30)=(l,  0,  — 2n,  0, 
where  on  the  left-hand  sides  I have,  for  greater  distinctness,  written  H,  &c.  to  denote 
the  functional  operation  of  taking  the  Hessian,  &c.  of  the  operand  aU+/30. 
In  particular,  if  u=0,  j3=l, 
HO=-n.H 
oo=-□^u 
□ 0=  n\ 
119.  Solution  of  a cubic  equation. 
The  question  is  to  find  a linear  factor  of  the  cubic 
{a,  b,  c,  dXx,  yy, 
and  this  can  be  at  once  effected  by  means  of  the  relation 
