464 
ME.  A.  CAYLEY’S  FIETH  l^IEMOIE  TPOY  QI7ANT4CS. 
143.  The  equation  1=0  gives 
A:B:C=l:a;:^y^ 
where  u is  an  imaginary  cube  root  of  unity ; the  factors  of  the  quartic  may  be  said  in 
this  case  to  be  Symmetric  Harmonics. 
The  equation  J = 0 gives  one  of  the  three  equations, 
A=B,  B=C,  C=A; 
in  this  case  a pair  of  factors  of  the  quartic  are  harmonics  with  respect  to  the  other  pau' 
of  factors.  If  we  have  simultaneously  1=0,  J=0,  then 
A=B=C=0, 
and  m this  case  three  of  the  factors  of  the  quartic  are  equal. 
144.  If  any  two  of  the  linear  factors  of  the  quartic  are  considered  as  forming,  with 
the  other  two  linear  factors,  an  involution,  the  sibiconjugates  of  the  involution  make  up 
a quadratic  factor  of  the  cubicovariant ; and  considering  the  thi’ee  pairs  of  sibiconju- 
gates, or  what  is  the  same  thing,  the  six  linear  factors  of  the  cubicovariant,  the  factors 
of  a pair  are  the  sibiconjugates  of  the  involution  formed  by  the  other  two  pau's  of 
factors. 
In  fact,  the  sibiconjugates  of  the  involution  formed  by  the  equations 
{w—ay){x—hj)  = 0,  {x- ^y){x—yy)  = 0 
are  found  by  means  of  the  Jacobian  of  these  two  functions,  riz.  of  the  quadrics 
(2,  — a,  yy 
(2,  -(i-y,  2(5yXx,  2j)\ 
which  is 
(^+a— /3— y,  —ha-\-(5y,  ^a((3-\-y)—(iy{^+a)Xx,y)-, 
viz.  a quadratic  factor  of  the  cubicovariant ; and  forming  the  other  two  factors,  there  is 
rro  difficulty  hr  seeirrg  that  any  one  of  these  is  the  Jacobiarr  of  the  other  two. 
145.  In  the  case  of  a pair  of  equal  roots,  we  have 
«~*U=  (x—ciyy(x—yy)(x—hy), 
ff-T  = iV(a— y)"(a— 
J = — 2 T 6 ( a — r )'( « — ^)^ 
□ = 0, 
a-^ILz=—^^-{2(a—yy(x—hyy+2(a  — '^y{x—yyy-jr{y—'^f{x—ciyy}(x—ayy-, 
— S)^(2a  — y — y^ — a**,  ya'^+^a'^ — 2ya^X^^  yT(^ — '^yY- 
In  the  case  of  two  pairs  of  equal  roots,  we  have 
«~‘U=  {x—ay)\x—yyf, 
a-^1  = -h(oc—yy, 
a ®J  = 
□ = 0, 
«-‘H=  —h{«‘-7)\x—ay)\x-yy)\ 
a>=  0; 
