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XXIII.  A Fifth  Memoir  upon  Quantics.  By  Aethue  Cayley,  Esp,  P.R.S. 
Eeeeived  February  11, — Bead  March  18,  1858. 
The  present  memoir  was  originally  intended  to  contain  a development  of  the  theories  of 
the  covariants  of  certain  binary  quantics,  viz.  the  quadric,  the  cubic,  and  the  quartic ; 
but  as  regards  the  theories  of  the  cubic  and  the  quartic,  it  was  found  necessary  to  con- 
sider the  case  of  two  or  more  quadrics,  and  I have  therefore  comprised  such  systems  of 
two  or  more  quadrics,  and  the  resulting  theories  of  the  harmonic  relation  and  of  invo- 
lution, in  the  subject  of  the  memoir ; and  although  the  theory  of  homography  or  of  the 
anharmonic  relation  belongs  rather  to  the  subject  of  bipartite  binary  quadrics,  yet  from 
its  connexion  with  the  theories  just  referred  to,  it  is  also  considered  in  the  memoir 
The  paragraphs  are  numbered  continuously  mth  those  of  my  former  memoirs  on  the 
subject:  Nos.  92  to  95  relate  to  a single  quadric;  Nos.  96  to  114  to  two  or  more  qua- 
drics, and  the  theories  above  referred  to ; Nos.  115  to  127  to  the  cubic,  and  Nos.  128 
to  145  to  the  quartic.  The  several  quantics  are  considered  as  expressed  not  only  in 
terms  of  the  coefficients,  but  also  in  terms  of  the  roots, — and  I consider  the  question  of 
the  determination  of  their  linear  factors, — a question,  in  effect,  identical  with  that  of 
the  solution  of  a quadric,  cubic,  or  biquadratic  equation.  The  expression  for  the  linear 
factor  of  a quadric  is  deduced  from  a well-known  formula ; those  for  the  linear  factors 
of  a cubic  and  a quartic  were  first  given  in  my  “ Note  sur  les  Covariants  d’une  fonction 
quadratique,  cubique  ou  biquadratique  a deux  indeterminees,”  Ceelle,  voL  L.  pp.  285 
to  287,  1855.  It  is  remarkable  that  they  are  in  one  point  of  view  more  simple  than 
the  expression  for  the  linear  factor  of  a quadric. 
92.  In  the  case  of  a quadric  the  expressions  considered  are 
(«,  h,  cjpc,  y)\  (1) 
ac-¥  , (2) 
where  (1)  is  the  quadric,  and  (2)  is  the  discriminant,  which  is  also  the  quadrin variant, 
catalecticant,  and  Hessian. 
And  where  it  is  convenient  to  do  so,  I write 
(1)  =U, 
(2)  =□. 
93.  We  have 
— dj,  yfn=V, 
which  expresses  that  the  evectant  of  the  discriminant  is  equal  to  the  quadric  ; 
