I^IE.  A.  CAYLEY’S  FIETH  MEMOIR  UPON  QUANTICS. 
443 
between  the  covariants.  The  equation  in  fact  shows  that  each  of  the  expressions 
i(<l>+uyo),  J(®-Un/d) 
is  a perfect  cube,  and  consequently  that  the  cube  root  of  each  of  these  expressions  is  a 
linear  function  of  (w,  y).  The  expression 
is  consequently  a linear  function  of  .r,  y,  and  it  vanishes  when  U = 0,  that  is,  the  expres- 
sion is  a linear  factor  of  the  cubic. 
It  may  be  noticed  here  that  the  cubic  being  a{x — ay){x—^y){x  — yy),  then  we  may 
write 
where  ai  is  an  imaginary  cube  root  of  unity : this  will  appear  from  the  expressions  which 
will  be  presently  given  for  the  covariants  in  terms  of  the  roots. 
120.  Canonical  form  of  the  cubic. 
The  expressions  + □),  — □)  are  perfect  cubes;  and  if  we  write 
i(4>+uyn)=ynx- 
i(<t._Uv'a)  = -v'Dy’, 
then  we  have 
U=x®+y^ 
0=yD(x^-y*), 
and  thence  also 
H=-^Dxy. 
121.  When  the  cubic  is  expressed  in  terms  of  the  roots,  we  have 
a ^V=(x-ciy)(x-(iy)(x-'yy) ; 
and  then  putting  for  shortness 
so  that 
we  have 
A=((3  — y)(x~ay), 
B=(y-cc)(x-f3y), 
C={c6-(i){x—yy), 
A-fB4-C=:0, 
a-^H  =— iV(A^+BHC=*)=i(BC+CA+AB), 
= -^(B-CXC- AXA-B), 
a-*0  = —^{(B— y)\y— af{cc—^)\ 
122.  The  covariants  H,  O are  most  simply  expressed  as  above,  but  it  may  be  proper 
to  add  the  equations 
