422 
ME.  A.  CAYLEY’S  FOLTITH  MEMOIE  TPOY  QrA^’TICS. 
coefficients,  that  is,  it  is  a cubinvariant.  This  agrees  with  the  before-mentioned  theorem 
for  the  number  of  cubinvariants. 
79.  In  the  case  of  the  quartic  («,  5,  c,  d,  ejjc,  y)^  the  cubinvariant  is,  by  the  preceding 
mode  of  generation,  obtained  in  the  form 
e{ac—h^) — id^{ad—hc) + Qc\{ae—^hd-\-  3c^)— 4^(5^—  cd)-\-a{ce—  d^), 
which  is  in  fact  equal  to 
?>{ace—ad^—lr‘e-{-2hcd—c^) ; 
and  omitting  the  numerical  factor  3,  we  have  the  cubinvariant  of  the  quartic. 
80.  In  the  case  of  a quantic  of  any  order  even  or  odd,  the  quadrin variants  of  the 
quadricovariants  are  quartinvariants  of  the  quantic.  But  these  quartinvaiiants  are  not  all 
of  them  independent,  and  there  is  no  obvious  method  grounded  on  the  preceding  mode 
of  generation  for  obtaining  the  number  of  the  independent  (asyzygetic)  quartinvaiiants. 
and  thence  the  number  of  the  irreducible  quartinvariants  of  a quantic  of  a given  order. 
81.  I take  the  opportunity  of  giving  some  additional  developments  in  relation  to  the 
discriminant  of  a quantic 
(a,  b,  ..  b\  i/)"’. 
To  render  the  signification  perfectly  definite,  it  should  be  remarked  that  the  discriminant 
contains  the  term  and  that  the  coefficient  of  this  term  may  be  taken  to  be  +1. 
It  was  noticed  in  the  introductory  memoir,  that,  by  Joachimsthal’s  theorem,  the  discri- 
minant, on  putting  «=0,  becomes  divisible  by  and  that  throwing  out  this  factor  it  is 
to  a numerical  factor  jrm  the  discriminant  of  the  quantic  of  the  order  (m— 1)  obtained 
by  putting  a=0  and  throwing  out  the  factor  ; and  it  was  also  remarked,  that  tliis  theo- 
rem, combined  with  the  general  property  of  invariants,  afforded  a convenient  method  for 
the  calculation  of  the  discriminant  of  a quantic  when  that  of  the  order  immediately  pre- 
ceding is  known.  Thus  let  it  be  proposed  to  find  the  discriminant  of  the  cubic 
(a,  b,  c,  dXx,  y)\ 
Imagine  the  discriminant  expanded  in  powers  of  the  leading  coefficient  a in  the  form 
Aa’^+Ba-l-C, 
then  this  function  qua  invariant  must  be  reduced  to  zero  by  the  operation  3Jd„+2cd4-l-(7d,; 
or  putting  for  shortness  V = the  operation  is  and  we  have 
and  consequently 
tt^VA-j-aVB  H-VC]_q 
-f«65A-f3iBj“  ’ 
A=-Vvb,  VA=0. 
But  C is  equal  to  into  the  discriminant  of  (3i,  3c,  dd^x,  yf,  that  is,  its  value  is 
h\\2bd—^(f),  or  throwing  out  the  factor  3,  we  may  write 
Cz=Wd-W(f-, 
