ME.  A.  CAYLEY’S  FOUETH  MEMOIE  LPON  QUANTICS. 
423 
this  gives 
or  reducing, 
and  thence 
B=  — 65c(:?+4c^; 
or  reducing,  A=d^, 
which  verifies  the  equation  VA=0,  and  the  discriminant  is,  as  we  know, 
a^d^ — 6abcd-\-  i.ac^  -\-Wd  — 35  V. 
82.  If  we  consider  the  quantic  (a,  b..a'yx,  I)”*  as  expressed  in  terms  of  the  roots  in 
the  form  a{x—ay)[x — (3^)...,  then  the  discriminant  &c.  as  above)  is  to  a 
factor  pres  equal  to  the  product  of  the  squares  of  the  difierences  of  the  roots,  and  the 
factor  may  be  determined  as  follows : viz.  denoting  by  l{a,  jS,  ...)  the  product  of  the 
squares  of  the  difierences  of  the  roots,  we  may  write 
j3,  &c.), 
where  N is  a number;  and  then  considering  the  equation  x’'‘—l  = 0,  we  have  to  deter- 
mine N the  equation 
But  in  general 
and  if 
then 
or 
here 
and  therefore 
but 
or 
and 
whence 
or 
and  consequently 
i:(c,  ,3,  ..)=(-)’«-N. 
/3..)=(-)^^'”-'^(a-/3)(a--y)..(/3— a)(/3— y)... 
^X=(x—a)(x—(3) . ., 
(c6—l3)(c6—y)..  = (p'a,  &c., 
(px=x”‘—l,  (p’x='mx”‘~\ 
<p'a(p'j3 . .=m”‘(c6(3y . . 
(-rccf3y..=  -l, 
^(3y..  = (-)-% 
<p'oi(p'/3 . . = ( — ; 
^(ce,  /3...)=(  — 
N 
13,  ..)=(  — &c.), 
3 L 
MDCCCLVIII. 
