ME.  A.  CAYLEY’S  FIFTH  MBMOIE  UPON  QUANTICS. 
445 
and  multiplying  this  by  (3  — y,  and  taking  the  sum  of  the  analogous  expressions,  this 
sum  vanishes,  or  the  three  pau’s  form  an  involution.  That  the  Hessian  gives  the  sibi- 
conjugates  of  the  involution  is  most  readily  shown  as  follows : — the  last-mentioned 
quadric  may  be  written 
( — (a + /3 + y)  + + 2 -f  C6y + /3y — a(a + /3 + + ( — 3a^y -1- a(a/3 + ay + ))/ , 
which  is  equal  to 
(3^+3.),r“+2(3^-3^.)*y+(3^+3^»y, 
or,  throwing  out  the  factor  3«~',  to 
{h+aoi,  2c— 25a,  d-\-coi'Jjic,  yf, 
which  is  harmonically  related  to  the  Hessian 
{ac—lf,  ad — he,  bd—c^^x,yY; 
and  in  like  manner  the  other  two  pairs  of  factors  will  be  also  harmonically  related  to 
the  Hessian. 
126.  In  the  case  of  a pair  of  equal  roots,  we  have 
cr‘U=  {x—aijf{x—yy), 
-^{cc—y)\x—ay)\ 
—^{ci—y)\x—ayy, 
□ = 0. 
And  in  the  case  of  all  the  roots  equal,  we  have 
=(x—ayy, 
H=0,  0=0,  0=0. 
127.  In  the  solution  of  a biquadratic  equation  we  have  to  consider  the  cubic  equation 
ro®— M(ot  — 1)  = 0.  The  cubic  here  is  (1,  0,  — M,  1)®,  or  what  is  the  same  thing, 
(1,  0, -pi,  MX^,  1)^ 
the  Hessian  is 
M(-i,  1, 1)- 
the  cubicovariant  is 
M(-l,  |M, -iM,  M+^M=“X^,  1)^ 
and  the  discriminant  is 
iVP(l-^M). 
128.  In  the  case  of  a quartic,  the  expressions  considered  are — 
{a,  h,  c,  d,  ejx,  y)*. 
(1) 
«c— 45fZ+3c^, 
(2) 
{ac—¥,  2{ad—hc),  ae-\-1hd—^& , 2(5c- 
-cd),  ce—d%x,  yf,  (3) 
ace  -}-  2hcd — ad^ — h^e — c^ 
(4) 
