452 
ME.  A.  CAYLEY’S  EIETH  IVIEMOIE  ETON  QTJAYTICS. 
whence  all  the  above-mentioned  determinants  will  vanish,  or  the  qnartic  will  be  a per- 
fect fourth  power  if  only  the  Hessian  vanishes  identically. 
138.  Considering  the  quartic  as  expressed  in  terms  of  the  roots,  we  have 
a-^V={x—uy){x—^y){x-yy){x-ly)’, 
and  if  we  write  for  shortness 
which  are  connected  by 
then  we  have 
A=(f3  — y)(ci—h), 
C = (a-^)(y-h), 
a+b+c=o, 
«-^I=^(A^+B^+C^)=-3L(bC+CA+AB), 
«-H=^(B-C)(C-AXA-B); 
and  for  the  discriminant  we  have 
□ = Tk(« — — y)X“ — — Wiv — 
=-2i6A^B^C^ 
and  it  is  easy  by  means  of  a preceding  formula  to  verify  the  equation  □ =1^  — 27.!'. 
139.  The  formulae  show  a very  remarkable  analogy  between  the  co variants  of  a cubic 
and  the  invariants  of  a quartic.  In  fact — 
For  the  cubic. 
B = (y—a)(x—(3y), 
C = (oi-(3)(x—yy), 
For  the  quartic. 
A=(/3  — yXa— ^), 
B = (y— aX/^  — ^), 
C = («-/3Xy-^). 
And  then  we  have  corresponding  to  each  other — 
For  the  cubic. 
The  Hessian, 
The  cubicovariant, 
The  cubic  into  the  square  root  of  the  discriminant. 
For  the  quartic. 
The  quadrinvariant, 
The  cubinvariant, 
The  discriminant. 
140.  For  the  two  covariants,  Ave  have 
«-^H=-^2(«- 
and 
if  for  shortness, 
(5y(x-yyf(x-hyf, 
y?  — ^a+/3y,  ^a(/3+7)— /3y(§-|-a)X^r,  y)", 
B = — y— a,  — ^/3-fya,  §/3(y +a)— ya(§+/3)Xa^, 
C=(S+y  — a— /3,  — ^y4-a/3,  ^y(a 4-/3)  — a/3(^-l-y )Xa^’,  yf. 
141.  We  have 
2 7 (A^B^-hCT  . 
8 (b-c)XC-a)XA-bx’ 
