446 
ME.  A.  CAYLEY’S  ELFTH  ^^lEilOIE  LTOY  QUA^TTICS. 
— a^d-\-  2)abc—  2h\,  ^ 
— (j^e  — ‘lab(i-\-  ^ac^  — ^h^c, 
— babe-\-lbacd—l%'^d^ 
<(  -^lOad^ —1%%  ^x,yf, 
+ bade-^-l^hd"^  — \bbce, 
+ ae^ -\-  2hde—  ^c^e-{-Qcd^, 
+ be^  — ?>cde-\-  2d^ 
(5) 
where  (1)  is  the  quartic,  (2)  is  the  quadrinvariant,  (3)  is  the  quadricoyariant  or  Hessian, 
(4)  is  the  cubin variant,  and  (6)  is  the  cubicovariant. 
And  where  it  is  convenient  to  do  so,  I write 
(1)  = U, 
(2)  = I , 
(3)  = H, 
(4)  = J, 
(5)  = O. 
The  preceding  covariants  are  connected  by  the  equation 
JU^  - lU^H + 4H^  = - 
The  discriminant  is  not  an  irreducible  invariant,  its  value  is 
□ — P— 27P=aV+  &c., 
for  which  see  Table  No.  12. 
129.  It  is  for  some  purposes  convenient  to  arrange  the  expanded  expression  of  the 
discriminant  in  powers  of  the  middle  coethcient  c.  We  thus  have 
□ = «V — Vla^bde^ — 2^o^d^ — Qab^d^e — 27^V— 645^(Z® 
+ c(5  4a^Pe + 5 4a  JV +108aJ(^^+10  Sb^de) 
-\-c\  — 18aV — lSOabde-\-^Qb^d^) 
+ — 5 4a6^^ — 5 4:b^e) 
-\-c*(Slae). 
130.  Solution  of  a biquadratic  equation. 
We  have  to  find  a linear  factor  of  the  quartic 
(a,  b,  c,  d,  eX^,  y)"- 
The  equation  JU^ — IU^H4-4H®= — <!>*,  putting  for  shortness 
13 
may  be  written 
(1,  0,  -M,  Wim,  JU)^=-iPO\ 
